$$
\begin{equation} \tag{1.1.5}
\begin{aligned}
E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\}) &= \{ (c_{n_\epsilon} \mu_{n_\epsilon}(\cdot))_\epsilon \in E(\mathbf{M}(\mathbb{R}^\infty, \Sigma)) \mid (\forall n \in \mathbb{N}) ((\forall A \in \Sigma(\mathbb{R}^\infty))) \\
&\qquad \quad (\exists p \in \mathbb{N}) [ |c_{n_\epsilon} \mu_{n_\epsilon}(A)| = O(\epsilon^{-p}) ] \}, \\
\mu_{n_\epsilon}(\cdot) &\in \mathbf{M}(\mathbb{R}^{n_\epsilon}, \Sigma_{n_\epsilon}), \Sigma_{n_\epsilon} = \mathbb{R}^{n_\epsilon} \cap \Sigma(\mathbb{R}^\infty), \\
|c_{n_\epsilon}| &\to \infty \text{ if } \epsilon \to 0,
\end{aligned}
\end{equation}
$$

$$
\begin{equation} \tag{1.1.6}
\begin{aligned}
N(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma), \{c_{n_{\epsilon}}\}) &= \{ (c_{n_{\epsilon}} \mu_{n_{\epsilon}}(\cdot))_{\epsilon} \in E(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma)) \mid (\forall n \in \mathbb{N}) ((\forall A \in \Sigma(\mathbb{R}^{\infty}))) \\
&\quad (\forall q \in \mathbb{N}) [ |c_{n_{\epsilon}} \mu_{n_{\epsilon}}(A)| = O(\epsilon^q) ] \},
\end{aligned}
\end{equation}
\[
\mu_{n_{\epsilon}}(\cdot) \in \mathbf{M}(\mathbb{R}^{n_{\epsilon}}, \Sigma_{n_{\epsilon}}), \Sigma_{n_{\epsilon}} = \mathbb{R}^{n_{\epsilon}} \cap \Sigma(\mathbb{R}^{\infty}).
\]
$$

$$
G(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma), \{c_{n_\epsilon}\}) = E_M(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma), \{c_{n_\epsilon}\})/N_M(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma), \{c_{n_\epsilon}\}). \quad (1.1.7)
$$

Elements of $E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ and $N_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ are called moderate, resp.negligible Normed by secuence $\{c_{n_\epsilon}\}$ Colombeau-Feynman measures or simply **Colombeau-Feynman measures**.

Note that $E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is an algebra with operations (i) $\forall c \in \mathbb{C}, \forall A \in \Sigma : ((c\mu_\epsilon)_\epsilon)(A) = c(\mu_\epsilon(A))_\epsilon$ (ii) $\forall A \in \Sigma : ((\mu_\epsilon + v_\epsilon)_\epsilon)(A) = (\mu_\epsilon(A))_\epsilon + (v_\epsilon(A))_\epsilon$.

It is the largest subalgebra of $E(\mathbf{M}(\mathbb{R}^\infty, \Sigma))$ in which $N_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is ideal.

Thus, $G(\mathbf{M}(\mathbb{R}^{\infty}, \Sigma), \{c_{n_{\epsilon}}\})$ is an associative, commutative algebra.

If $(\mu_\epsilon(\cdot))_\epsilon \in E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is a representative of $\mu \in G(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$, we write $\mu = \text{cl}[(\mu_\epsilon(\cdot))_\epsilon]$ or simply $\mu = [(\mu_\epsilon(\cdot))_\epsilon]$.

**Definition 1.1.5.** (1) For $\mathbb{R}$-valued functional $f \in (\mathbb{R}^\infty)^*$ we define the infinite dimensional upper integral $\overline{\mathfrak{I}}_\infty(f)$ as upper limit
$$E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\}) = \{ (c_{n_\epsilon} \mu_{n_\epsilon}(\cdot))_\epsilon \in E(\mathbf{M}(\mathbb{R}^\infty, \Sigma)) | (\forall n \in \mathbb{N}) ((\forall A \in \Sigma(\mathbb{R}^\infty))) $$

$$(\exists p \in \mathbb{N})[|c_{n_\epsilon} \mu_{n_\epsilon}(A)| = O(\epsilon^{-p})], \qquad (1.1.5)$$

$$\mu_{n_\epsilon}(\cdot) \in \mathbf{M}(\mathbb{R}^{n_\epsilon}, \Sigma_{n_\epsilon}), \Sigma_{n_\epsilon} = \mathbb{R}^{n_\epsilon} \cap \Sigma(\mathbb{R}^{\infty}),$$

$$|c_{n_\epsilon}| \to \infty \text{ if } \epsilon \to 0,$$

$$N(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\}) = \{ (c_{n_\epsilon} \mu_{n_\epsilon}(\cdot))_\epsilon \in E(\mathbf{M}(\mathbb{R}^\infty, \Sigma)) | (\forall n \in \mathbb{N}) ((\forall A \in \Sigma(\mathbb{R}^\infty))) $$

$$(\forall q \in \mathbb{N})[|c_{n_\epsilon} \mu_{n_\epsilon}(A)| = O(\epsilon^q)], \qquad (1.1.6)$$

$$\mu_{n_\epsilon}(\cdot) \in \mathbf{M}(\mathbb{R}^{n_\epsilon}, \Sigma_{n_\epsilon}), \Sigma_{n_\epsilon} = \mathbb{R}^{n_\epsilon} \cap \Sigma(\mathbb{R}^{\infty}).$$

$$G(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\}) = E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})/N_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\}). \qquad (1.1.7)$$

Elements of $E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ and $N_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ are called moderate, resp. negligible Normed by sequence $\{c_{n_\epsilon}\}$. Colombeau-Feynman measures or simply **Colombeau-Feynman measures**.

Note that $E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is an algebra with operations (i) $\forall c \in \mathbb{C}, \forall A \in \Sigma : ((c\mu_\epsilon)_\epsilon)(A) = c(\mu_\epsilon(A))_\epsilon$ (ii) $\forall A \in \Sigma : ((\mu_\epsilon + v_\epsilon)_\epsilon)(A) = (\mu_\epsilon(A))_\epsilon + (v_\epsilon(A))_\epsilon$.

It is the largest subalgebra of $E(\mathbf{M}(\mathbb{R}^\infty, \Sigma))$ in which $N_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is ideal.

Thus, $G(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is an associative, commutative algebra.

If $(\mu_\epsilon(\cdot))_\epsilon \in E_M(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$ is a representative of $\mu \in G(\mathbf{M}(\mathbb{R}^\infty, \Sigma), \{c_{n_\epsilon}\})$, we write $\mu = \text{cl}[(\mu_\epsilon(\cdot))_\epsilon]$ or simply $\mu = [(\mu_\epsilon(\cdot))_\epsilon]$.

**Definition 1.1.5.** (1) For $\mathbb{R}$-valued functional $f \in (\mathbb{R}^\infty)^*$ we define the infinite dimensional upper integral $\overline{\mathfrak{I}}_\infty(f)$ as upper limit
$$
\begin{equation} \tag{1.1.8}
\begin{aligned}
\bar{\mathfrak{F}}_{\infty}(f) &= \overline{\int_{\mathbb{R}^{\infty}} f(\mathbf{x}) D[\mathbf{x}]} = \limsup_{n \to \infty} \int_{\mathbb{R}^n} f_n(x) c_n d^n x = \\
&\qquad \limsup_{n \to \infty} \int \dots \int f_n(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n.
\end{aligned}
\end{equation}
$$

(2) For $\mathbb{R}$-valued functional $f \in (\mathbb{R}^{\infty})^*$ we define the infinite dimensional lower integral $\bar{\mathfrak{F}}_{\infty}(f)$ as lower limit

$$
\begin{equation} \tag{1.1.9}
\begin{aligned}
\bar{\mathfrak{F}}_{\infty}(f) &= \int_{\mathbb{R}^{\infty}} f(\mathbf{x}) D[\mathbf{x}] = \liminf_{n \to \infty} \int_{\mathbb{R}^{\infty}} f_n(x) c_n d^n x = \\
&\qquad \liminf_{n \to \infty} \int \dots \int f_n(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n.
\end{aligned}
\end{equation}
$$

**Proposition 1.1.1.** Assume that $f \in (\mathbb{R}^{\infty})^*$ is an $\mathbb{R}$-valued functional, then
$\bar{\mathfrak{F}}_{\infty}(f) \le \bar{\mathfrak{F}}_{\infty}(f)$ and

$$
\begin{equation} \tag{1.1.10}
\begin{aligned}
\bar{\mathfrak{F}}_{\infty}(f) &= \overline{\int_{\mathbb{R}^{\infty}}} f(\mathbf{x}) D[\mathbf{x}] = \lim_{k \to \infty} \left( \sup_{m \ge k} \int_{\mathbb{R}^m} f_m(x) c_m d^m x \right), \\
\bar{\mathfrak{F}}_{\infty}(f) &= \underline{\int_{\mathbb{R}^{\infty}}} f(\mathbf{x}) D[\mathbf{x}] = \lim_{k \to \infty} \left( \inf_{m \ge k} \int_{\mathbb{R}^m} f_m(x) c_m d^m x \right).
\end{aligned}
\end{equation}
$$

**Proposition 1.1.2.** Suppose that $\bar{\mathfrak{F}}_{\infty}(f) = \bar{\mathfrak{F}}_{\infty}(f) = a_f < \infty$. Then $\bar{\mathfrak{F}}_{\infty}(f) = a_f$.

**Proposition 1.1.3.** Assume that $f_1, f_2 \in (\mathbb{R}^{\infty})^*$ and $f_1(\mathbf{x}) \le f_2(\mathbf{x})$, then
$$
\begin{align}
\bar{\mathfrak{F}}_{\infty}(f) &= \overline{\int_{\mathbb{R}^{\infty}} f(\mathbf{x})D[\mathbf{x}]} = \limsup_{n \to \infty} \int_{\mathbb{R}^n} f_n(x)c_n d^n x = \nonumber \\
&\qquad \limsup_{n \to \infty} \int \dots \int f_n(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n. \tag{1.1.8}
\end{align}
$$

(2) For $\mathbb{R}$-valued functional $f \in (\mathbb{R}^{\infty})^*$ we define the infinite dimensional lower integral $\bar{\mathfrak{F}}_{\infty}(f)$ as lower limit

$$
\begin{equation}
\begin{aligned}
\bar{\mathfrak{F}}_{\infty}(f) &= \int_{\mathbb{R}^{\infty}} f(\mathbf{x}) D[\mathbf{x}] = \liminf_{n \to \infty} \int_{\mathbb{R}^{\infty}} f_n(\mathbf{x}) c_n d^n \mathbf{x} = \\
&\quad \liminf_{n \to \infty} \int \dots \int f_n(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n.
\end{aligned}
\tag{1.1.9}
\end{equation}
$$

**Proposition 1.1.1.** Assume that $f \in (\mathbb{R}^{\infty})^*$ is an $\mathbb{R}$-valued functional, then
$\bar{\mathfrak{F}}_{\infty}(f) \le \bar{\mathfrak{F}}_{\infty}(f)$ and

$$
\begin{align}
\bar{\mathfrak{F}}_{\infty}(f) &= \overline{\int_{\mathbb{R}^{\infty}} f(\mathbf{x}) D[\mathbf{x}]} = \lim_{k \to \infty} \left( \sup_{m \ge k} \int_{\mathbb{R}^m} f_m(x) c_m d^m x \right), \tag{1.1.10} \\
\bar{\mathfrak{F}}_{\infty}(f) &= \underline{\int_{\mathbb{R}^{\infty}} f(\mathbf{x}) D[\mathbf{x}]} = \lim_{k \to \infty} \left( \inf_{m \ge k} \int_{\mathbb{R}^m} f_m(x) c_m d^m x \right). \nonumber
\end{align}
$$

**Proposition 1.1.2.** Suppose that $\bar{\mathfrak{F}}_{\infty}(f) = \bar{\mathfrak{F}}_{\infty}(f) = a_f < \infty$. Then $\bar{\mathfrak{F}}_{\infty}(f) = a_f$.

**Proposition 1.1.3.** Assume that $f_1, f_2 \in (\mathbb{R}^{\infty})^*$ and $f_1(\mathbf{x}) \le f_2(\mathbf{x})$, then
$$ 
\begin{gathered}
\overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] \le \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}], \\
\underline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] \le \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}].
\end{gathered}
\qquad (1.1.11) $$

**Proposition 1.1.4.** Assume that $f_1, f_2 \in (\mathbb{R}^\infty)^*$ is an $\mathbb{R}$-valued functionals then

$$ 
\begin{align}
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}] \le \overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x}) + f_2(\mathbf{x})] D[\mathbf{x}] \le \nonumber \\
& \le \overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}], \tag{1.1.12} \\
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}] \ge \underline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x}) + f_2(\mathbf{x})] D[\mathbf{x}] \ge \nonumber \\
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}]. \nonumber
\end{align} $$

## Appendix II.1.2. Upper and Lower Infinite-Dimensional Colombeau-Feynman Subintegrals.

**Definition 1.2.1.** Let $\mathbb{R}^\infty$ be the space $\mathbb{R}^\infty = \bigoplus_{n=1}^\infty \mathbb{R}^n$. Set
$$ E(\mathbb{R}^\infty) = (\mathbf{C}^\infty(\mathbb{R}^\infty))^I, I = (0,1], $$
$$ 
\begin{gather*}
\overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] \le \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}], \tag{1.1.11} \\
\underline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] \le \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}].
\end{gather*}
$$

**Proposition 1.1.4.** Assume that $f_1, f_2 \in (\mathbb{R}^\infty)^*$ is an $\mathbb{R}$-valued functionals then

$$ 
\begin{align}
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}] \le \overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x}) + f_2(\mathbf{x})] D[\mathbf{x}] \le \nonumber \\
& \le \overline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}], \tag{1.1.12} \\
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \overline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}] \ge \underline{\int}_{\mathbb{R}^{\infty}} [f_1(\mathbf{x}) + f_2(\mathbf{x})] D[\mathbf{x}] \ge \nonumber \\
& \int_{\mathbb{R}^{\infty}} [f_1(\mathbf{x})] D[\mathbf{x}] + \underline{\int}_{\mathbb{R}^{\infty}} [f_2(\mathbf{x})] D[\mathbf{x}]. \nonumber
\end{align}
$$

## Appendix II.1.2. Upper and Lower Infinite-Dimensional Colombeau-Feynman Subintegrals.

**Definition 1.2.1.** Let $\mathbb{R}^\infty$ be the space $\mathbb{R}^\infty = \bigoplus_{n=1}^\infty \mathbb{R}^n$. Set
$$ E(\mathbb{R}^\infty) = (\mathbf{C}^\infty(\mathbb{R}^\infty))^I, I = (0,1], $$
$$E_M(\mathbb{R}^{\infty}) = \{(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E(\mathbb{R}^{\infty}) | (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))$$

$$(\forall K \in \mathbb{R}^n)(\forall \alpha \in \mathbb{N}_0^m)(\exists p \in \mathbb{N}) \left[ \sup_{x \in K} |\partial^\alpha f_{n,\varepsilon}(x)| = O(\varepsilon^{-p}) \right], \quad (1.2.1)$$

$$(f_{n,\varepsilon}(\mathbf{x}))_{\varepsilon} = (f_{\varepsilon}(\mathbf{x}) \upharpoonright \mathbb{R}^n)_{\varepsilon},$$

$$N_M(\mathbb{R}^{\infty}) = \{(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E(\mathbb{R}^{\infty}) | (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))$$

$$(\forall K \in \mathbb{R}^n)(\forall \alpha \in \mathbb{N}_0^m)(\forall q \in \mathbb{N}) \left[ \sup_{x \in K} |\partial^\alpha f_{n,\varepsilon}(x)| = O(\varepsilon^q) \right], \quad (1.2.2)$$

$$G(\mathbb{R}^{\infty}) = E_M(\mathbb{R}^{\infty})/N_M(\mathbb{R}^{\infty}). \quad (1.2.3)$$

Elements of $E_M(\mathbb{R}^{\infty})$ and $N_M(\mathbb{R}^{\infty})$ are called moderate, resp. negligible functionals. Note that $E_M(\mathbb{R}^{\infty})$ is a differential algebra with pointwise operations. It is the largest differential subalgebra of $E(\mathbb{R}^{\infty})$ in which $N_M(\mathbb{R}^{\infty})$ is a differential ideal. Thus, $G(\mathbb{R}^{\infty})$ is an associative, commutative differential algebra. If $(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E_M(\mathbb{R}^{\infty})$ is a representative of $f \in G(\mathbb{R}^{\infty})$, we write $f = \mathbf{cl}[(f_{\varepsilon}(\mathbf{x}))_{\varepsilon}]$ or simply $f = [(f_{\varepsilon}(\mathbf{x}))_{\varepsilon}]$.

**Definition 1.2.2.** For $(f_{\varepsilon})_{\varepsilon} \in G_{C^{\infty}(\mathbb{R}^{\infty})} = G(\mathbb{R}^{\infty})$ we define (by using Definition 1.1.1) the infinite dimensional Colombeau-Feinman integral $\mathfrak{I}_{\infty}(\mathbf{cl}[(f_{\varepsilon})_{\varepsilon}]) \triangleq \mathbf{cl}[(\mathfrak{I}_{\infty}(f_{\varepsilon}))_{\varepsilon}]$ by formula
$$E_M(\mathbb{R}^{\infty}) = \{(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E(\mathbb{R}^{\infty}) | (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))$$

$$(\forall K \in \mathbb{R}^n)(\forall \alpha \in \mathbb{N}_0^m)(\exists p \in \mathbb{N}) \left[ \sup_{x \in K} |\partial^\alpha f_{n,\varepsilon}(x)| = O(\varepsilon^{-p}) \right], \quad (1.2.1)$$

$$(f_{n,\varepsilon}(\mathbf{x}))_{\varepsilon} = (f_{\varepsilon}(\mathbf{x}) \upharpoonright \mathbb{R}^n)_{\varepsilon},$$

$$N_M(\mathbb{R}^{\infty}) = \{(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E(\mathbb{R}^{\infty}) | (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))$$

$$(\forall K \in \mathbb{R}^n)(\forall \alpha \in \mathbb{N}_0^m)(\forall q \in \mathbb{N}) \left[ \sup_{x \in K} |\partial^\alpha f_{n,\varepsilon}(x)| = O(\varepsilon^q) \right], \quad (1.2.2)$$

$$G(\mathbb{R}^{\infty}) = E_M(\mathbb{R}^{\infty})/N_M(\mathbb{R}^{\infty}). \quad (1.2.3)$$

Elements of $E_M(\mathbb{R}^{\infty})$ and $N_M(\mathbb{R}^{\infty})$ are called moderate, resp. negligible functionals. Note that $E_M(\mathbb{R}^{\infty})$ is a differential algebra with pointwise operations. It is the largest differential subalgebra of $E(\mathbb{R}^{\infty})$ in which $N_M(\mathbb{R}^{\infty})$ is a differential ideal. Thus, $G(\mathbb{R}^{\infty})$ is an associative, commutative differential algebra. If $(f_{\varepsilon}(\mathbf{x}))_{\varepsilon} \in E_M(\mathbb{R}^{\infty})$ is a representative of $f \in G(\mathbb{R}^{\infty})$, we write $f = \mathbf{cl}[(f_{\varepsilon}(\mathbf{x}))_{\varepsilon}]$ or simply $f = [(f_{\varepsilon}(\mathbf{x}))_{\varepsilon}]$.

**Definition 1.2.2.** For $(f_{\varepsilon})_{\varepsilon} \in G_{C^{\infty}(\mathbb{R}^{\infty})} = G(\mathbb{R}^{\infty})$ we define (by using Definition 1.1.1) the infinite dimensional Colombeau-Feinman integral $\mathfrak{I}_{\infty}(\mathbf{cl}[(f_{\varepsilon})_{\varepsilon}]) \triangleq \mathbf{cl}[(\mathfrak{I}_{\infty}(f_{\varepsilon}))_{\varepsilon}]$ by formula
$$ (\mathfrak{I}_{\infty}(f_{\varepsilon}))_{\varepsilon} = \left( \int_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} = \\ \left( \lim_{n \to \infty} \int_{\mathbb{R}^n} f_{n, \varepsilon}(\mathbf{x}) c_n d^n \mathbf{x} \right)_{\varepsilon} = \qquad (1.2.4) \\ \left( \lim_{n \to \infty} \int \dots \int f_{n, \varepsilon}(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n \right)_{\varepsilon} $$

iff $\forall \varepsilon \in (0, 1]$ limit in RHS of Eq.(1.2.4) exists.

**Remark 1.2.1.** We note that one can rewrite Definition 1.2.2 using completely Colombeau form.

**Definition 1.2.3.** For $f \in G(\mathbb{R}^{\infty})$ we define the infinite dimensional Colombeau-Feinman integral $\mathfrak{I}_{\infty}((f_{\delta})_{\delta})$, $\delta \in (0, 1]$ as Colombeau quantity such that

$$ (\mathfrak{I}_{\infty}(f_{\delta}))_{\delta} = \left( \int_{\mathbb{R}^{\infty}} f_{\delta}(\mathbf{x}) D[\mathbf{x}] \right)_{\delta \in (0,1]} = \tilde{\mathfrak{C}} \left( \left( \left( \int_{\mathbb{R}^{\infty}} f_{\delta}(\mathbf{x}) D[\mathbf{x}, \varepsilon] \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \\ \left( \left( \left( \int_{\mathbb{R}^{n_{\varepsilon}}} f_{n_{\varepsilon}, \delta}(\mathbf{x}) c_{n_{\varepsilon}} d^{n_{\varepsilon}} \mathbf{x} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \qquad (1.2.5) \\ ((c_{n_{\varepsilon}})_{\varepsilon}) \left( \left( \left( \int_{\mathbb{R}^{n_{\varepsilon}}} f_{n_{\varepsilon}, \delta}(\mathbf{x}) d^{n_{\varepsilon}} \mathbf{x} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \\ ((c_{n_{\varepsilon}})_{\varepsilon}) \left( \left( \int \dots \int f_{n_{\varepsilon}, \delta}(x_1, x_2, \dots, x_{n_{\varepsilon}}) dx_1 dx_2 \dots dx_{n_{\varepsilon}} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} $$
$$ (\mathfrak{I}_{\infty}(f_{\varepsilon}))_{\varepsilon} = \left( \int_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} = \\ \left( \lim_{n \to \infty} \int_{\mathbb{R}^n} f_{n,\varepsilon}(\mathbf{x}) c_n d^n \mathbf{x} \right)_{\varepsilon} = \qquad (1.2.4) \\ \left( \lim_{n \to \infty} \int \dots \int f_{n,\varepsilon}(x_1, x_2, \dots, x_n) c_n dx_1 dx_2 \dots dx_n \right)_{\varepsilon} $$

iff $\forall \varepsilon \in (0, 1]$ limit in RHS of Eq.(1.2.4) exists.

**Remark 1.2.1.** We note that one can rewrite Definition 1.2.2 using completely Colombeau form.

**Definition 1.2.3.** For $f \in G(\mathbb{R}^{\infty})$ we define the infinite dimensional Colombeau-Feinman integral $\mathfrak{I}_{\infty}((f_{\delta})_{\delta})$, $\delta \in (0, 1]$ as Colombeau quantity such that

$$ (\mathfrak{I}_{\infty}(f_{\delta}))_{\delta} = \left( \int_{\mathbb{R}^{\infty}} f_{\delta}(\mathbf{x}) D[\mathbf{x}] \right)_{\delta \in (0,1]} = \tilde{\mathfrak{C}} \left( \left( \left( \int_{\mathbb{R}^{\infty}} f_{\delta}(\mathbf{x}) D[\mathbf{x}, \varepsilon] \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \\ \left( \left( \left( \int_{\mathbb{R}^{n_{\varepsilon}}} f_{n_{\varepsilon},\delta}(\mathbf{x}) c_{n_{\varepsilon}} d^{n_{\varepsilon}} \mathbf{x} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \qquad (1.2.5) \\ ((c_{n_{\varepsilon}})_{\varepsilon}) \left( \left( \left( \int_{\mathbb{R}^{n_{\varepsilon}}} f_{n_{\varepsilon},\delta}(\mathbf{x}) d^{n_{\varepsilon}} \mathbf{x} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} \right) = \tilde{\mathfrak{C}} \\ ((c_{n_{\varepsilon}})_{\varepsilon}) \left( \left( \int \dots \int f_{n_{\varepsilon},\delta}(x_1, x_2, \dots, x_{n_{\varepsilon}}) dx_1 dx_2 \dots dx_{n_{\varepsilon}} \right)_{\varepsilon \in (0,1]} \right)_{\varepsilon \in (0,1]} $$
**Definition 1.2.4.** (1) For $\tilde{\mathbb{R}}$-valued functional $f \in G(\mathbb{R}^{\infty})$ we define the infinite dimensional Colombeau-Feinman upper integral $\overline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon})$ by formula

$$ \overline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) = \left( \overline{\int}_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} = \left( \limsup_{n \to \infty} \int_{\mathbb{R}^n} f_{n,\varepsilon}(x) c_n d^n x \right)_{\varepsilon} = \\ \left( \limsup_{n \to \infty} \int \dots \int f_{n,\varepsilon}(x_1,x_2,\dots,x_n) c_n dx_1 dx_2 \dots dx_n \right)_{\varepsilon}. \qquad (1.2.6) $$

(2) For $\tilde{\mathbb{R}}$-valued Colombeau generalized functional $f \in (\mathbb{R}^{\infty})^*$ we define the infinite dimensional lower integral $\underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon})$ by formula

$$ \underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) = \left( \int_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} = \left( \liminf_{n \to \infty} \int_{\mathbb{R}^{\infty}} f_{n,\varepsilon}(x) c_n d^n x \right)_{\varepsilon} = \\ \left( \liminf_{n \to \infty} \int \dots \int f_{n,\varepsilon}(x_1,x_2,\dots,x_n) c_n dx_1 dx_2 \dots dx_n \right)_{\varepsilon}. \qquad (1.2.7) $$

**Proposition 1.2.1.** Assume that $f \in G(\mathbb{R}^{\infty})$ is $\tilde{\mathbb{R}}$-valued Colombeau generalized functional, then $\underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) \leq_{\tilde{\mathbb{R}}} \overline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon})$ and

$$ \overline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) = \left( \overline{\int}_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} =_{\tilde{\mathbb{R}}} \left( \lim_{k \to \infty} \left( \sup_{m \ge k} \int_{\mathbb{R}^m} f_{m,\varepsilon}(x) c_m d^m x \right) \right)_{\varepsilon}, \qquad (1.2.8) \\ \underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) = \left( \underline{\int}_{\mathbb{R}^{\infty}} f_{\varepsilon}(\mathbf{x}) D[\mathbf{x}] \right)_{\varepsilon} =_{\tilde{\mathbb{R}}} \left( \lim_{k \to \infty} \left( \inf_{m \ge k} \int_{\mathbb{R}^m} f_{m,\varepsilon}(x) c_m d^m x \right) \right)_{\varepsilon}. $$

**Proposition 1.2.2.** Suppose that $\overline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) =_{\tilde{\mathbb{R}}} \underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) =_{\tilde{\mathbb{R}}} a((f_{\varepsilon})_{\varepsilon}) < \infty, a_f \in \tilde{\mathbb{R}}$.
Then $\underline{\mathfrak{F}}_{\infty}((f_{\varepsilon})_{\varepsilon}) =_{\tilde{\mathbb{R}}} a((f_{\varepsilon})_{\varepsilon})$.

**Proposition 1.2.3.** Assume that $f_1, f_2 \in G(\mathbb{R}^{\infty})$ is $\tilde{\mathbb{R}}$-valued Colombeau generalized
**Definition 1.2.4.** (1) For $\tilde{\mathbb{R}}$-valued functional $f \in G(\mathbb{R}^\infty)$ we define the infinite dimensional Colombeau-Feinman upper integral $\overline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon)$ by formula

$$ \overline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \left( \overline{\int}_{\mathbb{R}^\infty} f_\epsilon(\mathbf{x}) D[\mathbf{x}] \right)_\epsilon = \left( \limsup_{n \to \infty} \int_{\mathbb{R}^n} f_{n,\epsilon}(x) c_n d^n x \right)_\epsilon = \\ \left( \limsup_{n \to \infty} \int \dots \int f_{n,\epsilon}(x_1,x_2,\dots,x_n) c_n dx_1 dx_2 \dots dx_n \right)_\epsilon. \quad (1.2.6) $$

(2) For $\tilde{\mathbb{R}}$-valued Colombeau generalized functional $f \in (\mathbb{R}^\infty)^*$ we define the infinite dimensional lower integral $\underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon)$ by formula

$$ \underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \left( \int_{\mathbb{R}^\infty} f_\epsilon(\mathbf{x}) D[\mathbf{x}] \right)_\epsilon = \left( \liminf_{n \to \infty} \int_{\mathbb{R}^n} f_{n,\epsilon}(x) c_n d^n x \right)_\epsilon = \\ \left( \liminf_{n \to \infty} \int \dots \int f_{n,\epsilon}(x_1,x_2,\dots,x_n) c_n dx_1 dx_2 \dots dx_n \right)_\epsilon. \quad (1.2.7) $$

**Proposition 1.2.1.** Assume that $f \in G(\mathbb{R}^\infty)$ is $\tilde{\mathbb{R}}$-valued Colombeau generalized functional, then $\underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) \le \tilde{\mathbb{R}} \overline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon)$ and

$$ \overline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \left( \overline{\int}_{\mathbb{R}^\infty} f_\epsilon(\mathbf{x}) D[\mathbf{x}] \right)_\epsilon = \tilde{\mathbb{R}} \left( \lim_{k \to \infty} \left( \sup_{m \ge k} \int_{\mathbb{R}^m} f_{m,\epsilon}(x) c_m d^m x \right) \right)_\epsilon, \\ \underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \left( \int_{\mathbb{R}^\infty} f_\epsilon(\mathbf{x}) D[\mathbf{x}] \right)_\epsilon = \tilde{\mathbb{R}} \left( \lim_{k \to \infty} \left( \inf_{m \ge k} \int_{\mathbb{R}^m} f_{m,\epsilon}(x) c_m d^m x \right) \right)_\epsilon. \quad (1.2.8) $$

**Proposition 1.2.2.** Suppose that $\overline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \tilde{\mathbb{R}} \underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \tilde{\mathbb{R}} a((f_\epsilon)_\epsilon) < \infty, a_f \in \tilde{\mathbb{R}}$.
Then $\underline{\mathfrak{F}}_\infty((f_\epsilon)_\epsilon) = \tilde{\mathbb{R}} a((f_\epsilon)_\epsilon)$.

**Proposition 1.2.3.** Assume that $f_1, f_2 \in G(\mathbb{R}^\infty)$ is $\tilde{\mathbb{R}}$-valued Colombeau generalized
functionals and $f_1(\mathbf{x}) \leq_{\tilde{\mathbb{R}}} f_2(\mathbf{x})$, then

$$ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon}, \qquad (1.2.9) \\ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon}. $$

**Proposition 1.2.4.** Assume that $f_1, f_2 \in G(\mathbb{R}^\infty)$ is an $\tilde{\mathbb{R}}$-valued Colombeau generalized functionals then

$$ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} + \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x}) + f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \\ \leq_{\tilde{\mathbb{R}}} \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} + \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon}, \\ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} + \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \tilde{\geq} \qquad (1.2.10) \\ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x}) + f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} \tilde{\geq} \\ \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon} + \left( \overline{\int}_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}] \right)_{\varepsilon}. $$

## Appendix II.2.1. Colombeau Signed Measures
functionals and $f_1(\mathbf{x}) \leq_{\tilde{\mathbb{R}}} f_2(\mathbf{x})$, then

$$ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon}, \qquad (1.2.9) \\ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon}. $$

**Proposition 1.2.4.** Assume that $f_1, f_2 \in G(\mathbb{R}^\infty)$ is an $\tilde{\mathbb{R}}$-valued Colombeau generalized functionals then

$$ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} + \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \leq_{\tilde{\mathbb{R}}} \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x}) + f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \\ \leq_{\tilde{\mathbb{R}}} \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} + \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon}, \\ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} + \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \tilde{\mathbb{R}} \geq \qquad (1.2.10) \\ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x}) + f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} \tilde{\mathbb{R}} \geq \\ \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{1,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon} + \left( \overline{\int_{\mathbb{R}^{\infty}} [f_{2,\varepsilon}(\mathbf{x})] D[\mathbf{x}]} \right)_{\varepsilon}. $$

## Appendix II.2.1. Colombeau Signed Measures
In this section we introduced Colombeau type generalization of the notion of the signed measures. We also introduced Colombeau type generalization of the notion of the submeasures.

**Definition 2.1.1.** Suppose $\Sigma_X$ is an $\sigma$-algebra on a set $X$. A function $\mu : \to [-\infty, \infty]$ is called a signed measure on $\Sigma_X$, if it has the properties below:

$$ (1) \quad \mu(A) \le \infty, \forall A \in \Sigma_X, \\ (2) \quad \mu(A) \ge -\infty, \forall A \in \Sigma_X, \tag{2.1.1} $$

$$ (3) \quad \mu(\emptyset) = 0, $$

(4) For any pairwise disjoint sequence $\{A_n\}_{n=1}^\infty$, $A_n \subset \Sigma_X$ one has the equality

$$ (4) \quad \mu(\cup_{n=1}^\infty A_n) = \sum_{n=1}^\infty A_n. \tag{2.1.2} $$

Let us to use the term “honest” measure, for a measure in the usual sense.

**Definition 2.1.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $\mu$ be a signed measure on $\Sigma_X$ such that $\forall A \in \Sigma_X$:

$$ -\infty < \mu(A) < \infty \tag{2.1.3} $$

A signed measure $\mu(A)$ with property (2.2.1.3) is called finite.

**Theorem 2.1.1.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $\mu$ be a signed measure on $\Sigma_X$. Then there exist subsets $X^+, X^- \in \Sigma_X$, with the following properties:

(i) $X^+ \cap X^- = \emptyset, X^+ \cup X^- = X$;

(ii) the maps $\mu^\pm : \to [-\infty, \infty]$, defined by

$$ \mu^\pm(A) = \pm\mu(A \cap X^\pm), \forall A \in \Sigma_X, \tag{2.1.4} $$
In this section we introduced Colombeau type generalization of the notion of the signed measures. We also introduced Colombeau type generalization of the notion of the submeasures.

**Definition 2.1.1.** Suppose $\Sigma_X$ is an $\sigma$-algebra on a set $X$. A function $\mu : \to [-\infty, \infty]$ is called a signed measure on $\Sigma_X$, if it has the properties below:

$$ (1) \mu(A) \leq \infty, \forall A \in \Sigma_X, $$

$$ (2) \mu(A) \geq -\infty, \forall A \in \Sigma_X, \qquad (2.1.1) $$

$$ (3) \mu(\emptyset) = 0, $$

(4) For any pairwise disjoint sequence $\{A_n\}_{n=1}^{\infty}$, $A_n \subset \Sigma_X$ one has the equality

$$ (4) \mu(\cup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} A_n. \qquad (2.1.2) $$

Let us to use the term “honest” measure, for a measure in the usual sense.

**Definition 2.1.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $\mu$ be a signed measure on $\Sigma_X$ such that $\forall A \in \Sigma_X$:

$$ -\infty < \mu(A) < \infty \qquad (2.1.3) $$

A signed measure $\mu(A)$ with property (2.2.1.3) is called finite.

**Theorem 2.1.1.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $\mu$ be a signed measure on $\Sigma_X$. Then there exist subsets $X^+, X^- \in \Sigma_X$, with the following properties:

(i) $X^+ \cap X^- = \emptyset, X^+ \cup X^- = X$;

(ii) the maps $\mu^{\pm} : \to [-\infty, \infty]$, defined by

$$ \mu^{\pm}(A) = \pm \mu(A \cap X^{\pm}), \forall A \in \Sigma_X, \qquad (2.1.4) $$
are “honest” measures on $\Sigma_X$;

(iii) one of the measures $\mu^\pm(A)$ is finite, and one has the equality

$$ \mu = \mu^+ - \mu^- \qquad (2.1.5) $$

**Definition 2.1.3.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. A function $\mu : \Sigma_X \to \mathbb{C}$ is called a complex measure on $\Sigma_X$, if it is $\sigma$-additive in the sense that for any pairwise disjoint sequence $\{A_n\}_{n=1}^\infty, \forall n(A_n \in \Sigma_X)$, one has the equality

$$ (4) \quad \mu(\cup_{n=1}^\infty A_n) = \sum_{n=1}^\infty A_n. \qquad (2.1.6) $$

**Theorem 2.1.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $\mu$ be either a signed measure, or a complex measure on $\Sigma_X$. For every $A \in \Sigma_X$, we define

$$ |\mu|(A) = \sup_{\cup_{k=1}^{\infty} A_k = X} \left\{ \sum_{k=1}^{\infty} |\mu(A_k)| : \{A_n\}_{n=1}^{\infty} \text{pairwise disjoint} \right\}. \qquad (2.1.7) $$

Then the map $|\mu|(A) : \Sigma_X \to [0, \infty]$ is an “honest” measure on $\Sigma_X$.

**Definition 2.1.4.** The “honest” measure $|\mu|(A)$, defined by (2.2.1.7), is called the variation measure of $\mu(A)$. By construction, we have the inequality

$$ |\mu(A)| \le |\mu|(A). \qquad (2.1.8) $$

In the case of signed measures, the variation measure is also given by the following.

**Theorem 2.1.3.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $\mu$ be a signed measure on $\Sigma_X$. Then one has the equality

$$ |\mu| = \mu^+ + \mu^-, \qquad (2.1.9) $$

where $\mu = \mu^+ - \mu^-$ is the Hahn-Jordan decomposition of $\mu$.

**Definition 2.2.1.5.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let
are “honest” measures on $\Sigma_X$;

(iii) one of the measures $\mu^{\pm}(A)$ is finite, and one has the equality

$$ \mu = \mu^{+} - \mu^{-}. \qquad (2.1.5) $$

**Definition 2.1.3.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. A function $\mu : \Sigma_X \to \mathbb{C}$ is called a complex measure on $\Sigma_X$, if it is $\sigma$-additive in the sense that for any pairwise disjoint sequence $\{A_n\}_{n=1}^{\infty}$, $\forall n(A_n \in \Sigma_X)$, one has the equality

$$ (4) \quad \mu(\cup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} A_n. \qquad (2.1.6) $$

**Theorem 2.1.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $\mu$ be either a signed measure, or a complex measure on $\Sigma_X$. For every $A \in \Sigma_X$, we define

$$ |\mu|(A) = \sup_{\cup_{k=1}^{\infty} A_k = X} \left\{ \sum_{k=1}^{\infty} |\mu(A_k)| : \{A_n\}_{n=1}^{\infty} \text{pairwise disjoint} \right\}. \qquad (2.1.7) $$

Then the map $|\mu|(A) : \Sigma_X \to [0, \infty]$ is an “honest” measure on $\Sigma_X$.

**Definition 2.1.4.** The “honest” measure $|\mu|(A)$, defined by (2.2.1.7), is called the variation measure of $\mu(A)$. By construction, we have the inequality

$$ |\mu(A)| \le |\mu|(A). \qquad (2.1.8) $$

In the case of signed measures, the variation measure is also given by the following.

**Theorem 2.1.3.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $\mu$ be a signed measure on $\Sigma_X$. Then one has the equality

$$ |\mu| = \mu^{+} + \mu^{-}, \qquad (2.1.9) $$

where $\mu = \mu^{+} - \mu^{-}$ is the Hahn-Jordan decomposition of $\mu$.

**Definition 2.2.1.5.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let
$\mathbf{M}_{\mathbb{R}}(\Sigma_X)$ be a set of the all finite signed measures on $\Sigma_X$. Set
$E_{\mathbb{R}}(\Sigma_X) = (\mathbf{M}_{\mathbb{R}}(\Sigma_X))^I, I = (0,1]$,

$$
E_M^{\mathbb{R}}(\Sigma_X) =
\left\{
  \left(
    \mu_{\varepsilon})_{\varepsilon} \in E_{\mathbb{R}}(\Sigma_X) \middle|
    (\forall n \in \mathbb{N})((\forall m \in \mathbb{N})) (\exists p \in \mathbb{N})
    \left[
      \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^{-p})
    \right]
  \right\},
\quad (2.1.10)
$$

$$
N_M^{\mathbb{R}}(\Sigma_X) =
\left\{
  \left(
    \mu_{\varepsilon})_{\varepsilon} \in E_{\mathbb{R}}(\Sigma_X) \middle|
    (\forall n \in \mathbb{N})((\forall m \in \mathbb{N})) (\forall q \in \mathbb{N})
    \left[
      \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^q)
    \right]
  \right\},
\quad (2.1.11)
$$

$$
G_{\mathbb{R}}(\Sigma_X) = E_{\mathbb{R}}^{\mathbb{R}}(\Sigma_X) / N_{\mathbb{R}}^{\mathbb{R}}(\Sigma_X). \qquad (2.1.12)
$$

Elements of $E_M^{\mathbb{R}}(\Sigma_X)$ and $N_M^{\mathbb{R}}(\Sigma_X)$ are called moderate, resp. negligible signed measures. Note that $G_{\mathbb{R}}(\Sigma_X)$ is algebra with pointwise operations. It is the largest subalgebra of $E_M^{\mathbb{R}}(\Sigma_X)$ in which $N_M^{\mathbb{R}}(\Sigma_X)$ is ideal. Thus, $G_{\mathbb{R}}(\Sigma_X)$ is an associative, commutative algebra. If $(\mu_\varepsilon)_\varepsilon \in E_M^{\mathbb{R}}(\Sigma_X)$ is a representative of $\mu \in G_{\mathbb{R}}(\Sigma_X)$ we write
$\mu = \text{cl}[(\mu_\varepsilon)_\varepsilon]$ or simply
$\mu = [(\mu_\varepsilon)_\varepsilon]$.

**Definition 2.1.6.** An element $\mu \in G_{\mathbb{R}}(\Sigma_X)$ and any representative $(\mu_\varepsilon)_\varepsilon \in E_M^{\mathbb{R}}(\Sigma_X)$ of $\mu$ is called Colombeau signed measure on $\Sigma_X$.

**Definition 2.1.7.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let $\mathbf{M}_{\mathbb{C}}(\Sigma_X)$ be a set of the all complex finite measures on $\Sigma_X$. Set
$E_{\mathbb{C}}(\Sigma_X) = (\mathbf{M}_{\mathbb{C}}(\Sigma_X))^I, I = (0,1]$,

$$
E_M^{\mathbb{C}}(\Sigma_X) =
\left\{
  \left(
    \mu_{\varepsilon})_{\varepsilon} \in E_{\mathbb{C}}(\Sigma_X) \middle|
    (\forall n \in \mathbb{N})((\forall m \in \mathbb{N})) (\exists p \in \mathbb{N})
    \left[
      \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^{-p})
    \right]
  \right\},
\quad (2.1.13)
$$
$\mathbf{M}_{\mathbb{R}}(\Sigma_X)$ be a set of the all finite signed measures on $\Sigma_X$. Set
$E_{\mathbb{R}}(\Sigma_X) = (\mathbf{M}_{\mathbb{R}}(\Sigma_X))^I, I = (0,1]$,

$$
E_M^{\mathbb{R}}(\Sigma_X) = \\
\left\{ (\mu_\epsilon)_\epsilon \in E_\mathbb{R}(\Sigma_X) \middle| \begin{array}{@{}l@{}} (\forall n \in \mathbb{N})(\forall m \in \mathbb{N}))(\exists p \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_\epsilon(A)| = O(\epsilon^{-p}) \right] \end{array} \right\}, \tag{2.1.10}
$$

$$
N_M^{\mathbb{R}}(\Sigma_X) = \\
\left\{ (\mu_\epsilon)_\epsilon \in E_{\mathbb{R}}(\Sigma_X) \middle| (\forall n \in \mathbb{N})(\forall m \in \mathbb{N})(\forall q \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_\epsilon(A)| = O(\epsilon^q) \right] \right\}, \qquad (2.1.11)
$$

$$
G_{\mathbb{R}}(\Sigma_X) = E_{\mathbb{R}}^{\mathbb{R}}(\Sigma_X) / N_{\mathbb{R}}^{\mathbb{R}}(\Sigma_X). \qquad (2.1.12)
$$

Elements of $E_M^{\mathbb{R}}(\Sigma_X)$ and $N_M^{\mathbb{R}}(\Sigma_X)$ are called moderate, resp. negligible signed
measures. Note that $G_{\mathbb{R}}(\Sigma_X)$ is algebra with pointwise operations. It is the largest
subalgebra of $E_M^{\mathbb{R}}(\Sigma_X)$ in which $N_M^{\mathbb{R}}(\Sigma_X)$ is ideal. Thus, $G_{\mathbb{R}}(\Sigma_X)$ is an associative,
commutative algebra. If $(\mu_\epsilon)_\epsilon \in E_M^{\mathbb{R}}(\Sigma_X)$ is a representative of $\mu \in G_{\mathbb{R}}(\Sigma_X)$ we write
$\mu = \text{cl}[(\mu_\epsilon)_\epsilon]$ or simply
$\mu = [(\mu_\epsilon)_\epsilon]$.

**Definition 2.1.6.** An element $\mu \in G_{\mathbb{R}}(\Sigma_X)$ and any representative $(\mu_\epsilon)_\epsilon \in E_M^{\mathbb{R}}(\Sigma_X)$ of $\mu$ is called Colombeau signed measure on $\Sigma_X$.

**Definition 2.1.7.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let $\mathbf{M}_{\mathbb{C}}(\Sigma_X)$ be a set of the all complex finite measures on $\Sigma_X$. Set
$E_{\mathbb{C}}(\Sigma_X) = (\mathbf{M}_{\mathbb{C}}(\Sigma_X))^I, I = (0,1]$,

$$
E_M^{\mathbb{C}}(\Sigma_X) = \\
\left\{ (\mu_\epsilon)_\epsilon \in E_{\mathbb{C}}(\Sigma_X) \middle| (\forall n \in \mathbb{N})(\forall m \in \mathbb{N}))(\exists p \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_\epsilon(A)| = O(\epsilon^{-p}) \right] \right\}, \quad (2.1.13)
$$
$$ N_M^C(\Sigma_X) = \\ \left\{ (\mu_\epsilon)_\epsilon \in E_C(\Sigma_X) \middle| (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))(\forall q \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_\epsilon(A)| = O(\epsilon^q) \right] \right\}, \qquad (2.1.14) $$

$$ G_C(\Sigma_X) = E_M^C(\Sigma_X)/N_M^C(\Sigma_X). \qquad (2.1.15) $$

Elements of $E_M^C(\Sigma_X)$ and $N_M^C(\Sigma_X)$ are called moderate, resp. negligible complex measures.

Note that $G_R(\Sigma_X)$ is algebra with pointwise operations. It is the largest subalgebra of $E_M^R(\Sigma_X)$ in which $N_M^R(\Sigma_X)$ is ideal. Thus, $G_R(\Sigma_X)$ is an associative, commutative algebra. If $(\mu_\epsilon)_\epsilon \in E_M^R(\Sigma_X)$ is a representative of $\mu \in G_R(\Sigma_X)$ we write $\mu = \text{cl}[(\mu_\epsilon)_\epsilon]$ or simply

$$ \mu = [(\mu_\epsilon)_\epsilon]. $$

**Definition 2.1.8.** An element $\mu \in G_R(\Sigma_X)$ and any representative $(\mu_\epsilon)_\epsilon \in E_M^R(\Sigma_X)$ of $\mu$ is called Colombeau complex measure on $\Sigma_X$.

**Theorem 2.1.4.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $(\mu_\epsilon)_\epsilon$ be Colombeau signed measure on $\Sigma_X$. Then $\forall \epsilon \in (0, 1]$ there exist subsets $X_\epsilon^+, X_\epsilon^-$, with the following properties:

$$ (i) \quad \forall \epsilon(\epsilon \in (0,1]) [X_\epsilon^+ \cap X_\epsilon^- = \emptyset], \quad \forall \epsilon(\epsilon \in (0,1]) [X_\epsilon^+ \cup X_\epsilon^- = X]; $$

$$ (ii) \text{ the maps } (\mu_\epsilon^\pm)_\epsilon \to [-\infty, +\infty]^l \text{ and } \mu^\pm \to \tilde{\mathbb{R}}, \text{ defined by} $$

$$ \mu_{\epsilon}^{\pm}(A) = \pm \mu_{\epsilon}(A \cap X_{\epsilon}^{\pm}), \forall A \in \Sigma_{X}, \forall \epsilon \in (0,1] \qquad (2.1.16) $$

$$ \mu^{\pm}(A) = \text{cl}[(\mu_{\epsilon}^{\pm})_{\epsilon}(A)] $$

are “honest” Colombeau measures on $\Sigma_X$;

(iii) $\forall \epsilon \in (0,1]$ measures $\mu_\epsilon^\pm(A)$ is finite, and one has the equality

$$ (\mu_\epsilon)_\epsilon = (\mu_\epsilon^+)_{\epsilon} - (\mu_\epsilon^-)_{\epsilon}. \qquad (2.1.17) $$
$$ N_M^C(\Sigma_X) = \\ \left\{ (\mu_\epsilon)_\epsilon \in E_C(\Sigma_X) \middle| (\forall n \in \mathbb{N})((\forall m \in \mathbb{N}))(\forall q \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_\epsilon(A)| = O(\epsilon^q) \right] \right\}, \qquad (2.1.14) $$

$$ G_C(\Sigma_X) = E_M^C(\Sigma_X)/N_M^C(\Sigma_X). \qquad (2.1.15) $$

Elements of $E_M^C(\Sigma_X)$ and $N_M^C(\Sigma_X)$ are called moderate, resp. negligible complex measures.

Note that $G_R(\Sigma_X)$ is algebra with pointwise operations. It is the largest subalgebra of $E_M^R(\Sigma_X)$ in which $N_M^R(\Sigma_X)$ is ideal. Thus, $G_R(\Sigma_X)$ is an associative, commutative algebra. If $(\mu_\epsilon)_\epsilon \in E_M^R(\Sigma_X)$ is a representative of $\mu \in G_R(\Sigma_X)$ we write $\mu = \text{cl}[(\mu_\epsilon)_\epsilon]$ or simply

$$ \mu = [(\mu_\epsilon)_\epsilon]. $$

**Definition 2.1.8.** An element $\mu \in G_R(\Sigma_X)$ and any representative $(\mu_\epsilon)_\epsilon \in E_M^R(\Sigma_X)$ of $\mu$ is called Colombeau complex measure on $\Sigma_X$.

**Theorem 2.1.4.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$, and let $(\mu_\epsilon)_\epsilon$ be Colombeau signed measure on $\Sigma_X$. Then $\forall \epsilon \in (0, 1]$ there exist subsets $X_\epsilon^+, X_\epsilon^-$, with the following properties:

$$ (i) \quad \forall \epsilon(\epsilon \in (0,1]) [X_\epsilon^+ \cap X_\epsilon^- = \emptyset], \quad \forall \epsilon(\epsilon \in (0,1]) [X_\epsilon^+ \cup X_\epsilon^- = X]; $$

$$ (ii) \text{ the maps } (\mu_\epsilon^\pm)_\epsilon \to [-\infty, +\infty]^l \text{ and } \mu^\pm \to \tilde{\mathbb{R}}, \text{ defined by} $$

$$ \mu_{\epsilon}^{\pm}(A) = \pm \mu_{\epsilon}(A \cap X_{\epsilon}^{\pm}), \forall A \in \Sigma_{X}, \forall \epsilon \in (0,1] \qquad (2.1.16) $$

$$ \mu^{\pm}(A) = \text{cl}[(\mu_{\epsilon}^{\pm})_{\epsilon}(A)] $$

are “honest” Colombeau measures on $\Sigma_X$;

(iii) $\forall \epsilon \in (0,1]$ measures $\mu_\epsilon^\pm(A)$ is finite, and one has the equality

$$ (\mu_\epsilon)_\epsilon = (\mu_\epsilon^+)_{\epsilon} - (\mu_\epsilon^-)_{\epsilon}. \qquad (2.1.17) $$
**Theorem 2.1.5.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $(\mu_\varepsilon)_\varepsilon$ be either a signed Colombeau measure, or a complex Colombeau measure on $\Sigma_X$. For every $A \in \Sigma_X$, we define

$$ |(\mu_{\varepsilon})_{\varepsilon}|(A) = \left( \sup_{\bigcup_{k=1}^{\infty} A_k = X} \left\{ \sum_{k=1}^{\infty} |\mu_{\varepsilon}(A_k)| : \{A_n\}_{n=1}^{\infty} \text{pairwise disjoint} \right\} \right)_{\varepsilon}. \quad (2.1.18) $$

Then the map $|(\mu_\varepsilon)_\varepsilon|(A) : \Sigma_X \to [0, \infty]^l$ is an “honest” Colombeau measure on $\Sigma_X$.

**Definition 2.1.9.** The “honest” Colombeau measure $|(\mu_\varepsilon)_\varepsilon|(A)$, defined by (2.2.1.18), is called the variation of Colombeau measure $(\mu_\varepsilon)_\varepsilon(A)$. By construction, we have the inequality

$$ |(\mu_{\varepsilon}(A))_{\varepsilon}| \leq |(\mu_{\varepsilon}|(A))_{\varepsilon}|. \quad (2.1.19) $$

In the case of signed Colombeau measures, the variation measure is also given by the following.

**Theorem 2.1.6.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $(\mu_\varepsilon)_\varepsilon$ be a signed Colombeau measure on $\Sigma_X$. Then $\forall A \in \Sigma_X$ one has the equality

$$ |(\mu_{\varepsilon})_{\varepsilon}|(A) = ((\mu_{\varepsilon}^{+})_{\varepsilon})(A) + ((\mu_{\varepsilon}^{-})_{\varepsilon})(A), \quad (2.1.20) $$

where $\varepsilon \in (0, 1]$ and $\mu_\varepsilon = \mu_\varepsilon^+ - \mu_\varepsilon^-$ is the Hahn-Jordan decomposition of $\mu_\varepsilon$. We remind now that definitions.

**Definition 2.1.10.** The signum function $\operatorname{sgn}(x)$ of a real number $x \in \mathbb{R}$ is defined as follows:

$$ \operatorname{sgn}(x) = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \quad (2.1.21) $$

We note that any real number can be expressed as the product of its absolute
**Theorem 2.1.5.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $(\mu_\varepsilon)_\varepsilon$ be either a signed Colombeau measure, or a complex Colombeau measure on $\Sigma_X$. For every $A \in \Sigma_X$, we define

$$ |(\mu_{\varepsilon})_{\varepsilon}|(A) = \left( \sup_{\bigcup_{k=1}^{\infty} A_k = X} \left\{ \sum_{k=1}^{\infty} |\mu_{\varepsilon}(A_k)| : \{A_n\}_{n=1}^{\infty} \text{pairwise disjoint} \right\} \right)_{\varepsilon}. \quad (2.1.18) $$

Then the map $|(\mu_\varepsilon)_\varepsilon|(A) : \Sigma_X \to [0, \infty]^l$ is an "honest" Colombeau measure on $\Sigma_X$.

**Definition 2.1.9.** The "honest" Colombeau measure $|(\mu_\varepsilon)_\varepsilon|(A)$, defined by (2.2.1.18), is called the variation of Colombeau measure $(\mu_\varepsilon)_\varepsilon(A)$. By construction, we have the inequality

$$ |(\mu_{\varepsilon}(A))_{\varepsilon}| \leq |(\mu_{\varepsilon}|(A))_{\varepsilon}|. \quad (2.1.19) $$

In the case of signed Colombeau measures, the variation measure is also given by the following.

**Theorem 2.1.6.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$ and let $(\mu_\varepsilon)_\varepsilon$ be a signed Colombeau measure on $\Sigma_X$. Then $\forall A \in \Sigma_X$ one has the equality

$$ |(\mu_{\varepsilon})_{\varepsilon}|(A) = ((\mu_{\varepsilon}^{+})_{\varepsilon})(A) + ((\mu_{\varepsilon}^{-})_{\varepsilon})(A), \quad (2.1.20) $$

where $\varepsilon \in (0, 1]$ and $\mu_\varepsilon = \mu_\varepsilon^+ - \mu_\varepsilon^-$ is the Hahn-Jordan decomposition of $\mu_\varepsilon$. We remind now that definitions.

**Definition 2.1.10.** The signum function $\operatorname{sgn}(x)$ of a real number $x \in \mathbb{R}$ is defined as follows:

$$ \operatorname{sgn}(x) = \begin{cases} -1 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } x > 0 \end{cases} \quad (2.1.21) $$

We note that any real number can be expressed as the product of its absolute
value and its sign function:

$$x = |x|\operatorname{sgn}(x) \qquad (2.1.22)$$

**Definition 2.1.11.** Let $X$ be a non-empty set. (1) The positive part $f^+(x)$ of a real-valued function $f: X \to \mathbb{R}$ is defined via formula:

$$f^+(x) = \max\{f(x), 0\} = \begin{cases} f(x) & \text{if } f(x) > 0 \\ 0 & \text{otherwise} \end{cases} \qquad (2.1.23)$$

(2) The negative part $f^-(x)$ of a real-valued function $f: X \to \mathbb{R}$ is defined via formula:

$$\begin{align*} f^-(x) &= \max\{-f(x), 0\} = \\ &\quad -\min\{f(x), 0\} = \tag{2.1.24} \\ &\quad \begin{cases} -f(x) & \text{if } f(x) < 0 \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

**Remark.2.1.1.** Note that both $f^+(x)$ and $f^-(x)$ are non-negative functions. The function $f(x)$ can be expressed in terms of $f^+(x)$ and $f^-(x)$ as

$$f(x) = f^{+}(x) - f^{-}(x) \qquad (2.1.25)$$

Also note that
value and its sign function:

$$x = |x|\operatorname{sgn}(x) \qquad (2.1.22)$$

**Definition 2.1.11.** Let $X$ be a non-empty set. (1) The positive part $f^+(x)$ of a real-valued function $f: X \to \mathbb{R}$ is defined via formula:

$$f^+(x) = \max\{f(x), 0\} = \begin{cases} f(x) & \text{if } f(x) > 0 \\ 0 & \text{otherwise} \end{cases} \qquad (2.1.23)$$

(2) The negative part $f^-(x)$ of a real-valued function $f: X \to \mathbb{R}$ is defined via formula:

$$\begin{align*} f^-(x) &= \max\{-f(x), 0\} = \\ &\quad -\min\{f(x), 0\} = \tag{2.1.24} \\ &\quad \begin{cases} -f(x) & \text{if } f(x) < 0 \\ 0 & \text{otherwise} \end{cases} \end{align*}$$

**Remark.2.1.1.** Note that both $f^+(x)$ and $f^-(x)$ are non-negative functions. The function $f(x)$ can be expressed in terms of $f^+(x)$ and $f^-(x)$ as

$$f(x) = f^{+}(x) - f^{-}(x) \qquad (2.1.25)$$

Also note that
$$f^{+}(x) = |f(x)| \text{sgn}^{+}(f(x)), \qquad (2.1.26)$$

$$f^{-}(x) = |f(x)| \text{sgn}^{-}(f(x)).$$

Therefore from Eqs.(2.1.25)-(2.1.26) one obtain

$$f(x) = [\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] |f(x)|. \qquad (2.1.27)$$

**Remark.2.1.2.** Consider a nonnegative finite measure $\nu : \Sigma_X \to \mathbb{R}_+$ on the space $(X, \Sigma_X)$ and a measurable function $f : X \to \mathbb{R}$ such that $\int_A |f(x)|dv(x) < \infty$. Then, a finite signed measure $\mu_{\text{sg}}(A)$ is given by formula:

$$\begin{align*}
\mu_{\text{sg}}(A) &= \int_A f(x) dv(x) = \\
\int_A [f^+(x) - f^-(x)] dv(x) &= \tag{2.1.28} \\
\int_A f^+(x) dv(x) - \int_A f^-(x) dv(x) &= \\
\mu_{\text{sg}}^{+}(A) - \mu_{\text{sg}}^{-}(A).
\end{align*}$$

for all $A \in \Sigma_X$.

**Remark.2.1.3.** Substitution Eq.(2.1.27) into Eq.(2.1.28) gives
$$
\begin{align}
f^{+}(x) &= |f(x)| \mathbf{sgn}^{+}(f(x)), \tag{2.1.26} \\
f^{-}(x) &= |f(x)| \mathbf{sgn}^{-}(f(x)).
\end{align}
$$

Therefore from Eqs.(2.1.25)-(2.1.26) one obtain

$$
f(x) = [\mathbf{sgn}^{+}(f(x)) - \mathbf{sgn}^{-}(f(x))] |f(x)|. \quad (2.1.27)
$$

**Remark.2.1.2.** Consider a nonnegative finite measure $v : \Sigma_X \to \mathbb{R}_+$ on the space $(X, \Sigma_X)$ and a measurable function $f : X \to \mathbb{R}$ such that $\int_A |f(x)|dv(x) < \infty$. Then, a finite signed measure $\mu_{\text{sg}}(A)$ is given by formula:

$$
\begin{align*}
\mu_{\text{sg}}(A) &= \int_A f(x) dv(x) = \\
\int_A [f^+(x) - f^-(x)] dv(x) &= \\
\int_A f^+(x) dv(x) - \int_A f^-(x) dv(x) &= \tag{2.1.28}
\end{align*}
$$

$$
\int_A f^+(x) dv(x) - \int_A f^-(x) dv(x) =
$$

$$
\mu_{\text{sg}}^{+}(A) - \mu_{\text{sg}}^{-}(A).
$$

for all $A \in \Sigma_X$.

**Remark.2.1.3.** Substitution Eq.(2.1.27) into Eq.(2.1.28) gives
$$ \mu_{\text{sg}}(A) = \int_{A} f(x) dv(x) = $$

$$ \int_{A} [\mathbf{sgn}^{+}(f(x)) - \mathbf{sgn}^{-}(f(x))] |f(x)| dv(x) = \qquad (2.1.29) $$

$$ \int_{A} |f(x)| d\tilde{v}_{\text{sg}}(x), $$

where $d\tilde{v}_{\text{sg}}(x) = [\mathbf{sgn}^{+}(f(x)) - \mathbf{sgn}^{-}(f(x))]dv(x).$

**Example 2.1.1.** Freshnel signed measure $\mu_{Fr}$ on $\Sigma_X$. $X = \mathbb{R}^n, f(x) = \sin(x^2)$, $dv(x) = d^nx$.

$$ \mu_{Fr}(A) = \int_{A} \sin(x^2) d^nx = $$

$$ \int_{A} [\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))] |\sin(x^2)| d^nx = \qquad (2.1.30) $$

$$ \int_{A} |\sin(x^2)| d\tilde{v}_{\text{sg}}(x), $$

$$ d\tilde{v}_{\text{sg}}(x) = [\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))] d^nx. $$

**Remark 2.1.4.** Let us rewrite (2.1.29) in equivalent form
$$ \mu_{\text{sg}}(A) = \int_{A} f(x) dv(x) = $$

$$ \int_{A} [\mathbf{sgn}^{+}(f(x)) - \mathbf{sgn}^{-}(f(x))] |f(x)| dv(x) = \qquad (2.1.29) $$

$$ \int_{A} |f(x)| d\tilde{v}_{\text{sg}}(x), $$

$$ \text{where } d\tilde{v}_{\text{sg}}(x) = [\mathbf{sgn}^{+}(f(x)) - \mathbf{sgn}^{-}(f(x))] dv(x). $$

**Example 2.1.1.** Freshnel signed measure $\mu_{Fr}$ on $\Sigma_X$. $X = \mathbb{R}^n, f(x) = \sin(x^2)$, $dv(x) = d^nx$.

$$ \mu_{Fr}(A) = \int_{A} \sin(x^2) d^nx = $$

$$ \int_{A} [\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))] |\sin(x^2)| d^nx = \qquad (2.1.30) $$

$$ \int_{A} |\sin(x^2)| d\tilde{v}_{\text{sg}}(x), $$

$$ d\tilde{v}_{\text{sg}}(x) = [\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))] d^nx. $$

**Remark 2.1.4.** Let us rewrite (2.1.29) in equivalent form
$$ \mu_{\text{sg}}(A) = \int_{A} f(x) dv(x) = $$

$$ \int_{A} [\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] |f(x)| dv(x) = $$

$$ \int_{A} \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1 - 1\} |f(x)| dv(x) = \qquad (2.1.31) $$

$$ \int_{A} \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1\} |f(x)| dv(x) - \int_{A} |f(x)| dv(x) = $$

$$ \int_{A} |f(x)| d\tilde{v}_{+}(x) - \int_{A} |f(x)| dv(x), $$

where positive measure $d\tilde{v}_{+}(x)$ is given via formula

$$ d\tilde{v}_{+}(x) = \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1\} dv(x). \qquad (2.1.32) $$

**Example 2.1.2.** Freshnel signed measure $\mu_{Fr}$ on $\Sigma_X$. $X = \mathbb{R}^n$, $f(x) = \sin(x^2)$, $dv_+(x) = d^nx$.
$$ \mu_{\text{sg}}(A) = \int_{A} f(x) dv(x) = $$

$$ \int_{A} [\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] |f(x)| dv(x) = $$

$$ \int_{A} \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1 - 1\} |f(x)| dv(x) = \qquad (2.1.31) $$

$$ \int_{A} \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1\} |f(x)| dv(x) - \int_{A} |f(x)| dv(x) = $$

$$ \int_{A} |f(x)| d\tilde{v}_{+}(x) - \int_{A} |f(x)| dv(x), $$

where positive measure $d\tilde{v}_{+}(x)$ is given via formula

$$ d\tilde{v}_{+}(x) = \{[\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] + 1\} dv(x). \qquad (2.1.32) $$

**Example 2.1.2.** Freshnel signed measure $\mu_{Fr}$ on $\Sigma_X$. $X = \mathbb{R}^n$, $f(x) = \sin(x^2)$, $dv_+(x) = d^nx$.
$$ \mu_{Fr}(A) = \int_{A} \sin(x^2) d^n x = $$

$$ \int_{A} \{[\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))]+1-1\} |\sin(x^2)| d^n x = $$

$$ \int_{A} \{[\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))]+1\} |\sin(x^2)| d^n x - \int_{A} |\sin(x^2)| d^n x = (2.1.33) $$

$$ \int_{A} |\sin(x^2)| d\tilde{\nu}_{+}(x) - \int_{A} |\sin(x^2)| d^{n}x, $$

$$ d\tilde{\nu}_{+}(x) = \{[\mathbf{sgn}^{+}(\sin(x^2)) - \mathbf{sgn}^{-}(\sin(x^2))]+1\} d^{n}x. $$

**Definition 2.1.12.** Let $X$ be a non-empty set.

(1) The positive part $(f_{\varepsilon}(x))_{\varepsilon}^{+}$ of a real-valued generalized function $(f_{\varepsilon})_{\varepsilon}: X \to \tilde{\mathbb{R}}$ is defined via formula:

$$ (f_{\varepsilon}(x))_{\varepsilon}^{+} = (f_{\varepsilon}^{+}(x))_{\varepsilon}; \qquad (2.1.34) $$

(2) The negative part $(f_{\varepsilon}(x))_{\varepsilon}^{-}$ of a real-valued generalized function $(f_{\varepsilon})_{\varepsilon}: X \to \tilde{\mathbb{R}}$ is defined via formula

$$ (f_{\varepsilon}(x))_{\varepsilon}^{-} = (f_{\varepsilon}^{-}(x))_{\varepsilon}. \qquad (2.1.35) $$

**Remark.2.1.5.** We note that both $(f_{\varepsilon}^{+}(x))_{\varepsilon}$ and $(f_{\varepsilon}^{-}(x))_{\varepsilon}$ are non-negative. Colombeau generalized functions. The any Colombeau generalized function $(f_{\varepsilon}(x))_{\varepsilon}$ can be expressed in terms of $(f_{\varepsilon}(x))_{\varepsilon}^{+}$ and $(f_{\varepsilon}(x))_{\varepsilon}^{-}$ as
$$ \mu_{Fr}(A) = \int_A \sin(x^2) d^n x = $$

$$ \int_A \{[\mathbf{sgn}^+(\sin(x^2)) - \mathbf{sgn}^-(\sin(x^2))]+1-1\} |\sin(x^2)| d^n x = $$

$$ \int_A \{[\mathbf{sgn}^+(\sin(x^2)) - \mathbf{sgn}^-(\sin(x^2))]+1\} |\sin(x^2)| d^n x - \int_A |\sin(x^2)| d^n x = (2.1.33) $$

$$ \int_A |\sin(x^2)| d\tilde{\nu}_+(x) - \int_A |\sin(x^2)| d^n x, $$

$$ d\tilde{\nu}_+(x) = \{[\mathbf{sgn}^+(\sin(x^2)) - \mathbf{sgn}^-(\sin(x^2))]+1\} d^n x. $$

**Definition 2.1.12.** Let $X$ be a non-empty set.

(1) The positive part $(f_\varepsilon(x))^+ \varepsilon$ of a real-valued generalized function $(f_\varepsilon)_\varepsilon : X \to \tilde{\mathbb{R}}$ is defined via formula:

$$ (f_\varepsilon(x))^+ = (f_\varepsilon^+(x))_\varepsilon; \qquad (2.1.34) $$

(2) The negative part $(f_\varepsilon(x))^- \varepsilon$ of a real-valued generalized function $(f_\varepsilon)_\varepsilon : X \to \tilde{\mathbb{R}}$ is defined via formula

$$ (f_\varepsilon(x))_\varepsilon^- = (f_\varepsilon^-(x))_\varepsilon. \qquad (2.1.35) $$

**Remark.2.1.5.** We note that both $(f_\varepsilon^+(x))_\varepsilon$ and $(f_\varepsilon^-(x))_\varepsilon$ are non-negative. Colombeau generalized functions. The any Colombeau generalized function $(f_\varepsilon(x))_\varepsilon$ can be expressed in terms of $(f_\varepsilon(x))^+$ and $(f_\varepsilon(x))^-$ as
$$ (f_{\varepsilon}(x))_{\varepsilon} = (f_{\varepsilon}(x))_{\varepsilon}^{+} - (f_{\varepsilon}(x))_{\varepsilon}^{-}. \qquad (2.1.36) $$

Also we note that

$$ (f_{\varepsilon}(x))_{\varepsilon}^{+} = |(f_{\varepsilon}(x))_{\varepsilon}| \mathrm{sgn}^{+} [((f_{\varepsilon}(x)))_{\varepsilon}], \qquad (2.1.37) $$

$$ (f_{\varepsilon}(x))_{\varepsilon}^{-} = |(f_{\varepsilon}(x))_{\varepsilon}| \mathrm{sgn}^{-} [((f_{\varepsilon}(x)))_{\varepsilon}]. $$

Therefore from Eqs.(2.1.36)-(2.1.37) one obtain

$$ (f_{\varepsilon}(x))_{\varepsilon} = [\mathrm{sgn}^{+} [((f_{\varepsilon}(x)))_{\varepsilon}] - \mathrm{sgn}^{-} [((f_{\varepsilon}(x)))_{\varepsilon}]] |(f_{\varepsilon}(x))_{\varepsilon}|. \qquad (2.1.38) $$

**Remark.2.1.6.** Consider a nonnegative Colombeau measure $(v_{\varepsilon})_{\varepsilon} : \Sigma_X \to \tilde{\mathbb{R}}_+$ on the space $(X, \Sigma_X)$ and a measurable Colombeau generalized function $(f_{\varepsilon})_{\varepsilon} : X \to \tilde{\mathbb{R}}$ such that $\left( \int_A |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} < \infty$. Then, a finite signed Colombeau measure $(\mu_{\mathrm{sg}}(A, \varepsilon))_{\varepsilon}$ is given by formula:
$$ (f_{\varepsilon}(x))_{\varepsilon} = (f_{\varepsilon}(x))_{\varepsilon}^{+} - (f_{\varepsilon}(x))_{\varepsilon}^{-}. \qquad (2.1.36) $$

Also we note that

$$ (f_{\varepsilon}(x))_{\varepsilon}^{+} = |(f_{\varepsilon}(x))_{\varepsilon}| \mathrm{sgn}^{+} [((f_{\varepsilon}(x)))_{\varepsilon}], \qquad (2.1.37) $$

$$ (f_{\varepsilon}(x))_{\varepsilon}^{-} = |(f_{\varepsilon}(x))_{\varepsilon}| \mathrm{sgn}^{-} [((f_{\varepsilon}(x)))_{\varepsilon}]. $$

Therefore from Eqs.(2.1.36)-(2.1.37) one obtain

$$ (f_{\varepsilon}(x))_{\varepsilon} = [\mathrm{sgn}^{+} [((f_{\varepsilon}(x)))_{\varepsilon}] - \mathrm{sgn}^{-} [((f_{\varepsilon}(x)))_{\varepsilon}]] |(f_{\varepsilon}(x))_{\varepsilon}|. \qquad (2.1.38) $$

**Remark.2.1.6.** Consider a nonnegative Colombeau measure $(v_{\varepsilon})_{\varepsilon} : \Sigma_X \to \tilde{\mathbb{R}}_+$ on the space $(X, \Sigma_X)$ and a measurable Colombeau generalized function $(f_{\varepsilon})_{\varepsilon} : X \to \tilde{\mathbb{R}}$ such that $\left( \int_A |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} < \infty$. Then, a finite signed Colombeau measure $(\mu_{\mathrm{sg}}(A, \varepsilon))_{\varepsilon}$ is given by formula:
$$ (\mu_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} [f_{\varepsilon}^{+}(x) - f_{\varepsilon}^{-}(x)] d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \qquad (2.1.39) $$

$$ \left( \int_A f_{\varepsilon}^{+}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} - \left( \int_A f_{\varepsilon}^{-}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \\ (\mu_{\text{sg}}^{+}(A, \varepsilon))_{\varepsilon} - (\mu_{\text{sg}}^{-}(A, \varepsilon))_{\varepsilon}. $$

for all $A \in \Sigma_X$.

**Remark.2.1.7.** Substitution Eq.(2.1.38) into Eq.(2.1.39) gives

$$ \mu_{\text{sg}}(A, \varepsilon) = \int_{A} f_{\varepsilon}(x) d\nu_{\varepsilon}(x) = $$

$$ \left( \int_A \left[ \text{sgn}^+(f_\varepsilon(x)) - \text{sgn}^-(f_\varepsilon(x)) \right] |f_\varepsilon(x)| d\nu_\varepsilon(x) \right)_\varepsilon = \qquad (2.1.40) $$

$$ \left( \int_A |f_\varepsilon(x)| d\tilde{v}_{\text{sg}}(x, \varepsilon) \right)_\varepsilon, $$

where

$$ (d\tilde{v}_{\text{sg}}(x, \varepsilon))_{\varepsilon} = \left[ \text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x)) \right] d\nu(x). \qquad (2.1.41) $$
$$ (\mu_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} [f_{\varepsilon}^{+}(x) - f_{\varepsilon}^{-}(x)] d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \qquad (2.1.39) $$

$$ \left( \int_A f_{\varepsilon}^{+}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} - \left( \int_A f_{\varepsilon}^{-}(x) d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \\ (\mu_{\text{sg}}^{+}(A, \varepsilon))_{\varepsilon} - (\mu_{\text{sg}}^{-}(A, \varepsilon))_{\varepsilon}. $$

for all $A \in \Sigma_X$.

**Remark.2.1.7.** Substitution Eq.(2.1.38) into Eq.(2.1.39) gives

$$ \mu_{\text{sg}}(A, \varepsilon) = \int_{A} f_{\varepsilon}(x) d\nu_{\varepsilon}(x) = \\ \left( \int_{A} \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] |f_{\varepsilon}(x)| d\nu_{\varepsilon}(x) \right)_{\varepsilon} = \qquad (2.1.40) \\ \left( \int_{A} |f_{\varepsilon}(x)| d\tilde{v}_{\text{sg}}(x, \varepsilon) \right)_{\varepsilon}, $$

where

$$ (d\tilde{v}_{\text{sg}}(x, \varepsilon))_{\varepsilon} = [\text{sgn}^{+}(f(x)) - \text{sgn}^{-}(f(x))] d\nu(x). \qquad (2.1.41) $$
**Remark.2.1.8.** Let us rewrite (2.1.40) in equivalent form

$$ (\mu_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} - \\ \left( \int_{A} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} |f_{\varepsilon}(x)| d\tilde{\nu}_{\varepsilon}^{+}(x) \right)_{\varepsilon} - \left( \int_{A} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon}, $$

where positive Colombeau measure $(d\tilde{\nu}_{\varepsilon}^{+}(x))_{\varepsilon}$ is given via formula

$$ (d\tilde{\nu}_{\varepsilon}^{+}(x))_{\varepsilon} = \left( \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} dv_{\varepsilon}(x) \right)_{\varepsilon}. \quad (2.1.43) $$

**Definition 2.1.13.** Let $(\lambda_{\varepsilon})_{\varepsilon}$ and $(v_{\varepsilon})_{\varepsilon}$ be signed Colombeau measures on $(X, \Sigma_X)$ then $(\lambda_{\varepsilon})_{\varepsilon}$ is called absolutely continuous with respect to $(v_{\varepsilon})_{\varepsilon}$ ($(\lambda_{\varepsilon})_{\varepsilon} \ll (v_{\varepsilon})_{\varepsilon}$) if $(|v_{\varepsilon}|(A))_{\varepsilon} = 0$ imply $(\lambda_{\varepsilon}(A))_{\varepsilon} = 0$.

**Theorem 2.1.8.** Let $(\mu_{\varepsilon})_{\varepsilon}$ be $\sigma$-additive Colombeau measure on the space
**Remark.2.1.8.** Let us rewrite (2.1.40) in equivalent form

$$ (\mu_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} - \\ \left( \int_{A} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon} = \\ \left( \int_{A} |f_{\varepsilon}(x)| d\tilde{\nu}_{\varepsilon}^{+}(x) \right)_{\varepsilon} - \left( \int_{A} |f_{\varepsilon}(x)| dv_{\varepsilon}(x) \right)_{\varepsilon}, $$

where positive Colombeau measure $(d\tilde{\nu}_{\varepsilon}^{+}(x))_{\varepsilon}$ is given via formula

$$ (d\tilde{\nu}_{\varepsilon}^{+}(x))_{\varepsilon} = \left( \left\{ \left[ \text{sgn}^{+}(f_{\varepsilon}(x)) - \text{sgn}^{-}(f_{\varepsilon}(x)) \right] + 1 \right\} dv_{\varepsilon}(x) \right)_{\varepsilon}. \quad (2.1.43) $$

**Definition 2.1.13.** Let $(\lambda_{\varepsilon})_{\varepsilon}$ and $(v_{\varepsilon})_{\varepsilon}$ be signed Colombeau measures on $(X, \Sigma_X)$ then $(\lambda_{\varepsilon})_{\varepsilon}$ is called absolutely continuous with respect to $(v_{\varepsilon})_{\varepsilon}$ ($(\lambda_{\varepsilon})_{\varepsilon} \ll (v_{\varepsilon})_{\varepsilon}$) if $(|v_{\varepsilon}|(A))_{\varepsilon} = 0$ imply $(\lambda_{\varepsilon}(A))_{\varepsilon} = 0$.

**Theorem 2.1.8.** Let $(\mu_{\varepsilon})_{\varepsilon}$ be $\sigma$-additive Colombeau measure on the space
$(X, \Sigma_X), (v_{\text{sg}}(\cdot, \varepsilon))_{\varepsilon}$ be a signed Colombeau measure on $(X, \Sigma_X)$ and $(v_{\text{sg}}(\cdot, \varepsilon))_{\varepsilon}$ is absolutely continuous with respect to $(\mu_{\varepsilon})_{\varepsilon}$. Then there exists unique $(f_{\varepsilon})_{\varepsilon} \in G_{L_1(X, \mu)}$ such that

$$ (v_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) d\mu_{\varepsilon} \right)_{\varepsilon} \qquad (2.1.44) $$

for any $A \in \Sigma_X$.

## Appendix II.2.2. Colombeau Submeasures.

**Definition 2.2.1.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. A submeasure $\mu$ on $\Sigma_X$ is a function $\mu : \Sigma_X \to [0, +\infty]$ which has the properties $\forall A, B \in \Sigma_X$:

$$ \mu(\emptyset) = 0, $$

$$ A \subseteq B \Rightarrow \mu(A) \leq \mu(B), \qquad (2.2.1) $$

$$ \mu(A \cup B) \leq \mu(A) + \mu(B). $$

**Definition 2.2.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let $\text{SM}(\Sigma_X)$ be a set of the all finite submeasures on $\Sigma_X$. Set $\text{SE}(\Sigma_X) = (\text{SM}(\Sigma_X))^I, I = (0, 1]$,

$$ \text{SE}_M(\Sigma_X) = $$

$$ \left\{ (\mu_{\varepsilon})_{\varepsilon} \in \text{SE}(\Sigma_X) \middle| (\forall n \in \mathbb{N})((\forall m \in \mathbb{N})) (\exists p \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^{-p}) \right] \right\}, \qquad (2.2.2) $$
$(X, \Sigma_X), (v_{\text{sg}}(\cdot, \varepsilon))_{\varepsilon}$ be a signed Colombeau measure on $(X, \Sigma_X)$ and $(v_{\text{sg}}(\cdot, \varepsilon))_{\varepsilon}$ is absolutely continuous with respect to $(\mu_{\varepsilon})_{\varepsilon}$. Then there exists unique $(f_{\varepsilon})_{\varepsilon} \in G_{L_1(X, \mu)}$ such that

$$ (v_{\text{sg}}(A, \varepsilon))_{\varepsilon} = \left( \int_{A} f_{\varepsilon}(x) d\mu_{\varepsilon} \right)_{\varepsilon} \qquad (2.1.44) $$

for any $A \in \Sigma_X$.

## Appendix II.2.2. Colombeau Submeasures.

**Definition 2.2.1.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. A submeasure $\mu$ on $\Sigma_X$ is a function $\mu : \Sigma_X \to [0, +\infty]$ which has the properties $\forall A, B \in \Sigma_X$:

$$ \mu(\emptyset) = 0, $$

$$ A \subseteq B \Rightarrow \mu(A) \leq \mu(B), \qquad (2.2.1) $$

$$ \mu(A \cup B) \leq \mu(A) + \mu(B). $$

**Definition 2.2.2.** Let $X$ be a non-empty set, let $\Sigma_X$ be a $\sigma$-algebra on $X$. Let $\text{SM}(\Sigma_X)$ be a set of the all finite submeasures on $\Sigma_X$. Set $\text{SE}(\Sigma_X) = (\text{SM}(\Sigma_X))^{\prime}, I = (0, 1]$,

$$ \text{SE}_{M}(\Sigma_{X}) = $$

$$ \left\{ (\mu_{\varepsilon})_{\varepsilon} \in \text{SE}(\Sigma_X) \middle| (\forall n \in \mathbb{N})((\forall m \in \mathbb{N})) (\exists p \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^{-p}) \right] \right\}, \qquad (2.2.2) $$
$SN_M(\Sigma_X) =$

$$ \left\{ (\mu_{\varepsilon})_{\varepsilon} \in SE(\Sigma_X) \middle| (\forall n \in \mathbb{N}) \left( (\forall m \in \mathbb{N}) (\forall q \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^q) \right] \right) \right\}, \quad (2.2.3) $$

$$ SG(\Sigma_X) = SE_M(\Sigma_X)/SN_M(\Sigma_X). \quad (2.2.4) $$

Elements of $SE_M(\Sigma_X)$ and $SN_M(\Sigma_X)$ are called moderate, resp. negligible submeasures.

Note that $SG(\Sigma_X)$ is algebra with pointwise operations. It is the largest subalgebra of $SE_M(\Sigma_X)$ in which $SN_M(\Sigma_X)$ is ideal. Thus, $SG(\Sigma_X)$ is an associative, commutative algebra. If $(\mu_\varepsilon)_\varepsilon \in SE_M(\Sigma_X)$ is a representative of $\mu \in SG(\Sigma_X)$ we write $\mu = \text{cl}[(\mu_\varepsilon)_\varepsilon]$ or simply $\mu = [(\mu_\varepsilon)_\varepsilon]$.

**Definition 2.2.3.** An element $\mu \in SG(\Sigma_X)$ and any representative $(\mu_\varepsilon)_\varepsilon \in SE_M(\Sigma_X)$ of $\mu$ is called Colombeau submeasure on $\Sigma_X$.

**Definition 2.2.4.** Let $\mathbb{R}$ be a ring of subsets of a set $\mathfrak{J} \neq \emptyset$.

A set function $\mu : \mathbb{R} \to [0, \infty)$ is said to be a Dobrakov submeasure or D-submeasure, if it is

(1) monotone: if $E, F \in \mathbb{R}$ such that $E \subset F$, then $\mu(E) \le \mu(F)$;

(2) subadditively continuous: for every $F \in \mathbb{R}$ and $\varepsilon > 0$ there exists a $\delta > 0$ such that for every $E \in \mathbb{R}$ with $\mu(E) < \delta$

$$ (i) \mu(E \cup F) \le \mu(F) + \varepsilon, \quad (2.2.5) $$

$$ (ii) \mu(F) \le \mu(F \setminus E) + \varepsilon; $$

## Appendix II.3.1. Colombeau Signed Measures and Submeasures on Infinite-Dimensional Linear
$SN_M(\Sigma_X) =$

$$ \left\{ (\mu_{\varepsilon})_{\varepsilon} \in SE(\Sigma_X) \middle| (\forall n \in \mathbb{N}) \left( (\forall m \in \mathbb{N}) (\forall q \in \mathbb{N}) \left[ \sup_{A \in \Sigma_X} |\mu_{\varepsilon}(A)| = O(\varepsilon^q) \right] \right) \right\}, \quad (2.2.3) $$

$$ SG(\Sigma_X) = SE_M(\Sigma_X)/SN_M(\Sigma_X). \quad (2.2.4) $$

Elements of $SE_M(\Sigma_X)$ and $SN_M(\Sigma_X)$ are called moderate, resp. negligible submeasures.

Note that $SG(\Sigma_X)$ is algebra with pointwise operations. It is the largest subalgebra of $SE_M(\Sigma_X)$ in which $SN_M(\Sigma_X)$ is ideal. Thus, $SG(\Sigma_X)$ is an associative, commutative algebra. If $(\mu_\varepsilon)_\varepsilon \in SE_M(\Sigma_X)$ is a representative of $\mu \in SG(\Sigma_X)$ we write $\mu = \text{cl}[(\mu_\varepsilon)_\varepsilon]$ or simply $\mu = [(\mu_\varepsilon)_\varepsilon]$.

**Definition 2.2.3.** An element $\mu \in SG(\Sigma_X)$ and any representative $(\mu_\varepsilon)_\varepsilon \in SE_M(\Sigma_X)$ of $\mu$ is called Colombeau submeasure on $\Sigma_X$.

**Definition 2.2.4.** Let $\mathbb{R}$ be a ring of subsets of a set $\mathfrak{J} \neq \emptyset$.

A set function $\mu : \mathbb{R} \to [0, \infty)$ is said to be a Dobrakov submeasure or D-submeasure, if it is

(1) monotone: if $E, F \in \mathbb{R}$ such that $E \subset F$, then $\mu(E) \le \mu(F)$;

(2) subadditively continuous: for every $F \in \mathbb{R}$ and $\varepsilon > 0$ there exists a $\delta > 0$ such that for every $E \in \mathbb{R}$ with $\mu(E) < \delta$

$$ (i) \mu(E \cup F) \le \mu(F) + \varepsilon, \quad (2.2.5) $$

$$ (ii) \mu(F) \le \mu(F \setminus E) + \varepsilon; $$

## Appendix II.3.1. Colombeau Signed Measures and Submeasures on Infinite-Dimensional Linear
# Spaces.

In this section we introduced a generalization of the notion of the positive Feynman-Colombeau "measure" to the case where the values are allowed to be outside set $[0, \infty]$. We also introduced an infinite-dimensional generalization of the notion of the submeasures.

**Definition 2.2.2.1.** Suppose $\mathbb{R}^\infty$ is an $\sigma$-algebra on a set $\mathbb{R}^\infty = \bigoplus_{n=1}^\infty \mathbb{R}^n$. A function $\mu : \to [-\infty, \infty]$ is called a signed measure on $\mathbb{R}^\infty$, if it has the properties below:

$$ (1) \mu(A) \le \infty, \forall A \in \mathbb{R}^\infty, $$

$$ (2) \mu(A) \ge -\infty, \forall A \in \mathbb{R}^\infty, \qquad (2.2.1) $$

$$ (3) \mu(\emptyset) = 0, $$

(4) For any pairwise disjoint sequence $\{A_n\}_{n=1}^\infty$, $A_n \subset \mathbb{R}^\infty$ one has the equality

$$ (4) \mu(\cup_{n=1}^\infty A_n) = \sum_{n=1}^\infty A_n. \qquad (2.2.2) $$

**Examples 2.2.1.** Let us agree to use the term “honest” measure, for a measure in the usual sense.

1. Any “honest” measure on $\mathbb{R}^\infty$ is a signed measure on $\mathbb{R}^\infty$.

2. If $\mu$ is a signed measure on $\mathbb{R}^\infty$, then $-\mu$ is again a signed measure on $\mathbb{R}^\infty$.

3. If $\mu_1$ and $\mu_2$ are “honest” measures on $\mathbb{R}^\infty$, one of which is finite, then $\mu_1 - \mu_2$ is a signed measure on $\mathbb{R}^\infty$.

**Definition 2.2.2.** Let $M(\mathbb{R}^\infty)$ be a set of the all signed measures on $\mathbb{R}^\infty$.

Set $E(\mathbb{R}^\infty) = (M(\mathbb{R}^\infty))^I, I = (0,1]$,

**Definition 2.**(1) A submeasure $\mu$ on $\mathbb{R}^\infty$ is a setfunction which has the properties
Spaces.

In this section we introduced a generalization of the notion of the positive
Feynman-Colombeau "measure" to the case where the values are allowed to be
outside set [0, ∞]. We also introduced an infinite-dimensional generalization of the
notion of the submeasures.

**Definition 2.2.2.1.** Suppose $\mathbb{R}^{\infty}$ is an $\sigma$-algebra on a set $\mathbb{R}^{\infty} = \bigoplus_{n=1}^{\infty} \mathbb{R}^n$. A function $\mu : \to [-\infty, \infty]$ is called a signed measure on $\mathbb{R}^{\infty}$, if it has the properties below:

$$
(1) \mu(A) \le \infty, \forall A \in \mathbb{R}^{\infty},
$$

$$
(2) \mu(A) \ge -\infty, \forall A \in \mathbb{R}^{\infty}, \qquad (2.2.1)
$$

$$
(3) \mu(\emptyset) = 0,
$$

(4) For any pairwise disjoint sequence {$A_n$}$_{n=1}^\infty$, $A_n \subset \mathbb{R}^\infty$ one has the equality

$$
(4) \mu(\bigcup_{n=1}^{\infty} A_n) = \sum_{n=1}^{\infty} A_n. \qquad (2.2.2)
$$

**Examples 2.2.1.** Let us agree to use the term “honest” measure, for a measure in the usual sense.

1. Any "honest" measure on $\mathbb{R}^\infty$ is a signed measure on $\mathbb{R}^\infty$.

2. If $\mu$ is a signed measure on $\mathbb{R}^{\infty}$, then $-\mu$ is again a signed measure on $\mathbb{R}^{\infty}$.

3.If μ₁ and μ₂ are “honest” measures on ℝ∞, one of which is finite, then μ₁ – μ₂ is a signed measure on ℝ∞.

**Definition 2.2.2.** Let $M(\mathbb{R}^{\infty})$ be a set of the all signed measures on $\mathbb{R}^{\infty}$.

Set $E(\mathbb{R}^{\infty}) = (M(\mathbb{R}^{\infty}))^{I}, I = (0,1]$,

**Definition 2.**(1) A submeasure $\mu$ on $\mathbb{R}^\infty$ is a setfunction which has the properties
$$ \mu(\emptyset) = 0, $$

$$ A \subseteq B \Rightarrow \mu(A) \leq \mu(B), \qquad (2.2.) $$

$$ \mu(A \cup B) \leq \mu(A) + \mu(B). $$

## Appendix III. Colombeau-Albeverio Path Integral.

Let $\mathbf{H}$ be a real separable Hilbert space, considered as a measurable Hilbert space with the o-algebra generated by its open subsets. Let $\|\cdot\|$ ( be the norm in $\mathbf{H}$, We want to define a normalized subintegral on $\mathbf{H}$ of the following font

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon \in (0,1]} \qquad (3.1) $$

where $i$ is the imaginary unit, $(\Phi_{\varepsilon})_{\varepsilon}$ belongs to a suitable Colombeau class of "integrable functions" on $\mathbf{H}$ and $\sim$ above the integral reminds us to the normalization. Normalization should mean that

$$ (\mathbf{J}[1_{\varepsilon}])_{\varepsilon \in (0,1]} = \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon \in (0,1]} = \mathbf{c} \mathbf{l} [(1_{\varepsilon})_{\varepsilon \in (0,1]}]. \qquad (3.2) $$

To see what this implies let us took first to the case where $\mathbf{H}$ - is finite i.e. $\mathbf{H} = \mathbb{R}^d$.

Thus one obtain
$$ \mu(\emptyset) = 0, $$

$$ A \subseteq B \Rightarrow \mu(A) \leq \mu(B), \qquad (2.2.) $$

$$ \mu(A \cup B) \leq \mu(A) + \mu(B). $$

## Appendix III. Colombeau-Albeverio Path Integral.

Let $\mathbf{H}$ be a real separable Hilbert space, considered as a measurable Hilbert space with the o-algebra generated by its open subsets. Let $\|\cdot\|$ ( be the norm in $\mathbf{H}$, We want to define a normalized subintegral on $\mathbf{H}$ of the following font

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon \in (0,1]} \qquad (3.1) $$

where $i$ is the imaginary unit, $(\Phi_{\varepsilon})_{\varepsilon}$ belongs to a suitable Colombeau class of "integrable functions" on $\mathbf{H}$ and $\sim$ above the integral reminds us to the normalization. Normalization should mean that

$$ (\mathbf{J}[1_{\varepsilon}])_{\varepsilon \in (0,1]} = \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon \in (0,1]} = \mathbf{c} \mathbf{l} [(1_{\varepsilon})_{\varepsilon \in (0,1]}]. \qquad (3.2) $$

To see what this implies let us took first to the case where $\mathbf{H}$ - is finite i.e. $\mathbf{H} = \mathbb{R}^d$.

Thus one obtain
$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] D[\gamma] = 1, \qquad (3.3) $$

$$ i^{-\frac{d}{2}} = \exp\left(\frac{i\pi d}{4}\right). $$

Hence from Eq.(3.3),Eq.(3.2) we see that

$$ (\mathbf{J}[1_{\varepsilon}])_{\varepsilon \in (0,1]} = \left( \int_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] D[\gamma] \right)_{\varepsilon \in (0,1]} = \qquad (3.4) $$

$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] d^d\gamma. $$

We also want that the normalized subintegral given via Eq.(3.1) be translation Invariant, which is formally expressed by the writing $D[\gamma]$ in Eq.(3.4) as a Lebesgue measure $d^d\gamma$; in fact when $\mathbf{H} = \mathbb{R}^d$ the translation Invariance of the Lebesgue measure $d^d\gamma$ implies the translation invariance of the usual integral

$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] d^d\gamma. $$

The condition for translation invariance is

$$ \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma + \alpha_{\varepsilon}\|^2 \right] D[\gamma] \right)_{\varepsilon} = (1)_{\varepsilon}. \qquad (3.5) $$

for all $[(\alpha_{\varepsilon})_{\varepsilon}] \in G_{\mathbf{H}}$ i.e. $(\alpha_{\varepsilon})_{\varepsilon} \in E_M(\mathbf{H})$. Let now $\langle \cdot, \cdot \rangle$ be the scaler product in $\mathbf{H}$.
Then $\|\gamma + \alpha\|^2 = \|\gamma\|^2 + \langle 2\gamma, \alpha \rangle + \|\alpha\|^2, \varepsilon \in (0,1]$ and hence Eq.(3.5) is equivalent with

$$ \left( \int_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp[i\langle\gamma, \alpha\rangle] \exp\left[\frac{i}{2} \|\alpha\|^2\right] D[\gamma] \right)_{\varepsilon} = (1)_{\varepsilon} \qquad (3.6) $$
$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] D[\gamma] = 1, \qquad (3.3) $$

$$ i^{-\frac{d}{2}} = \exp\left(\frac{i\pi d}{4}\right). $$

Hence from Eq.(3.3),Eq.(3.2) we see that

$$ (\mathbf{J}[1_{\varepsilon}])_{\varepsilon \in (0,1]} = \left( \int_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] D[\gamma] \right)_{\varepsilon \in (0,1]} = \qquad (3.4) $$

$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] d^d\gamma. $$

We also want that the normalized subintegral given via Eq.(3.1) be translation Invariant, which is formally expressed by the writing $D[\gamma]$ in Eq.(3.4) as a Lebesgue measure $d^d\gamma$; in fact when $\mathbf{H} = \mathbb{R}^d$ the translation Invariance of the Lebesgue measure $d^d\gamma$ implies the translation invariance of the usual integral

$$ (2\pi i)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \exp\left[\frac{i}{2} \|\gamma\|^2\right] d^d\gamma. $$

The condition for translation invariance is

$$ \left( \int_{\mathbf{H}} \exp\left[ \frac{i}{2} \|\gamma + \alpha_{\varepsilon}\|^2 \right] D[\gamma] \right)_{\varepsilon} = (1)_{\varepsilon}. \qquad (3.5) $$

for all $[(\alpha_{\varepsilon})_{\varepsilon}] \in G_{\mathbf{H}}$ i.e. $(\alpha_{\varepsilon})_{\varepsilon} \in E_M(\mathbf{H})$. Let now $\langle \cdot, \cdot \rangle$ be the scaler product in $\mathbf{H}$.
Then $\|\gamma + \alpha\|^2 = \|\gamma\|^2 + \langle 2\gamma, \alpha \rangle + \|\alpha\|^2, \varepsilon \in (0,1]$ and hence Eq.(3.5) is equivalent with

$$ \left( \int_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp[i\langle\gamma, \alpha\rangle] \exp\left[\frac{i}{2} \|\alpha\|^2\right] D[\gamma] \right)_{\varepsilon} = (1)_{\varepsilon} \qquad (3.6) $$
i.e. observing that $\exp[\frac{i}{2} \|\alpha\|^2]$ is independent of the variable $\gamma$ we get, with the obvious postulate that the normalized subintegral be $\tilde{\mathbb{C}}$-linear:

$$ \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] D[\gamma] \right)_{\varepsilon} = \left( \exp\left[-\frac{i}{2} \|\alpha\|^2\right] \right)_{\varepsilon}. \qquad (3.7) $$

Set now

$$ (\Phi_{\alpha_{\varepsilon}}(\gamma))_{\varepsilon} = (\exp[i\langle\gamma, \alpha_{\varepsilon}\rangle])_{\varepsilon} \qquad (3.8) $$

and

$$ (\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\alpha_{\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon}. \qquad (3.9) $$

Then from Eq.(3.7) - Eq.(3.9) we get

$$ \begin{aligned} (\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon} &= \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\alpha_{\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon} \\ &= \left( \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] \right)_{\varepsilon}. \end{aligned} \qquad (3.10) $$

Eq.(3.10) gives us the evaluation of the functional $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ for the value of the argument $\Phi_{\alpha_{\varepsilon}}$. Extending now naturally the definition of $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ to linear combinations of functions (3.8), where $\alpha_{\varepsilon}$ varies over $E_M(\mathbf{H})$, and observing that when $\alpha_{\varepsilon}$ is the zero vector in $\mathbf{H}$ for all $\varepsilon \in (0,1]$ we have $(\Phi_{\alpha_{\varepsilon}})_{\varepsilon} = (1)_{\varepsilon}$ and $(\mathbf{J}[1])_{\varepsilon} = (1)_{\varepsilon}$, by Eq.(3.10) (which is in this case precisely the normalization condition (3.2)), we see that $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ becomes a normalized $\tilde{\mathbb{C}}$-linear functional on the linear vector space over $\tilde{\mathbb{C}}$ generated by the functions (3.8) (which is actually
i.e. observing that $\exp[\frac{i}{2} \|\alpha\|^2]$ is independent of the variable $\gamma$ we get, with the obvious postulate that the normalized subintegral be $\tilde{\mathbb{C}}$- linear:

$$ \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] D[\gamma] \right)_{\varepsilon} = \left( \exp\left[-\frac{i}{2} \|\alpha\|^2\right] \right)_{\varepsilon}. \qquad (3.7) $$

Set now

$$ (\Phi_{\alpha_{\varepsilon}}(\gamma))_{\varepsilon} = (\exp[i\langle\gamma, \alpha_{\varepsilon}\rangle])_{\varepsilon} \qquad (3.8) $$

and

$$ (\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\alpha_{\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon}. \qquad (3.9) $$

Then from Eq.(3.7) - Eq.(3.9) we get

$$ \begin{aligned} (\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon} &= \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\alpha_{\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon} \\ &= \left( \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] \right)_{\varepsilon}. \end{aligned} \qquad (3.10) $$

Eq.(3.10) gives us the evaluation of the functional $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ for the value of the argument $\Phi_{\alpha_{\varepsilon}}$. Extending now naturally the definition of $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ to linear combinations of functions (3.8), where $\alpha_{\varepsilon}$ varies over $E_M(\mathbf{H})$, and observing that when $\alpha_{\varepsilon}$ is the zero vector in $\mathbf{H}$ for all $\varepsilon \in (0,1]$ we have $(\Phi_{\alpha_{\varepsilon}})_{\varepsilon} = (1)_{\varepsilon}$ and $(\mathbf{J}[1])_{\varepsilon} = (1)_{\varepsilon}$, by Eq.(3.10) (which is in this case precisely the normalization condition (3.2)), we see that $(\mathbf{J}[\Phi_{\alpha_{\varepsilon}}])_{\varepsilon}$ becomes a normalized $\tilde{\mathbb{C}}$-linear functional on the linear vector space over $\tilde{\mathbb{C}}$ generated by the functions (3.8) (which is actually
identical with the algebra generated by such functions). We have then

$$ 
\begin{aligned}
\left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \sum_n \Phi_{\alpha_{n,\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon} &= \left( \mathbf{J} \left[ \sum_n \Phi_{\alpha_{n,\varepsilon}}(\gamma) \right] \right)_{\varepsilon} = \\
&\qquad \sum_n c_n \left( \exp\left[-\frac{i}{2} \|\alpha_{n,\varepsilon}\|^2\right] \right)_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] \sum_n \delta(\alpha_{\varepsilon} - \alpha_{n,\varepsilon}) D[\gamma] \right)_{\varepsilon},
\end{aligned}
\quad (3.11) $$

where $\delta(\alpha_{\varepsilon} - \alpha_{n,\varepsilon})$ is the measure concentrated at the origin in $\mathbf{H}$. Observing that

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \sum_{n} (\Phi_{\alpha_{n,\varepsilon}}(\gamma))_{\varepsilon} \quad (3.11') $$

is the Fourier-Colombeau transform on $\mathbf{H}$ of the Colombeau measure $(d\mu_{\Phi_{\varepsilon}}(\gamma))_{\varepsilon}$ i.e.

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.12) $$

we can rewrite Eq.(3.11) in the form

$$ 
\begin{aligned}
(\mathbf{J}[\Phi_{\varepsilon}(\gamma)])_{\varepsilon} &= \\
&\quad \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon}.
\end{aligned}
\quad (3.13) $$

We now observe that the right hand side of Eq.(3.13) is well defined also when $\mu_{\Phi_{\varepsilon}}$ is replaced by any bounded for all $\varepsilon \in (0,1]$ Colombeau complex measure on the measurable space $\mathbf{H}$, $\exp[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2]$ being a bounded continuous function. Since any such Colombeau measure $(\mu_{\varepsilon})_{\varepsilon}$ has a Colombeau-Fourier transform on $\mathbf{H}$,
identical with the algebra generated by such functions). We have then

$$ \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \sum_n \Phi_{\alpha_{n,\varepsilon}}(\gamma) D[\gamma] \right)_{\varepsilon} = \left( \mathbf{J} \left[ \sum_n \Phi_{\alpha_{n,\varepsilon}}(\gamma) \right] \right)_{\varepsilon} = \\ \sum_n c_n \left( \exp\left[-\frac{i}{2} \|\alpha_{n,\varepsilon}\|^2\right] \right)_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] \sum_n \delta(\alpha_{\varepsilon} - \alpha_{n,\varepsilon}) D[\gamma] \right)_{\varepsilon}, \quad (3.11) $$

where $\delta(\alpha_{\varepsilon} - \alpha_{n,\varepsilon})$ is the measure concentrated at the origin in $\mathbf{H}$. Observing that

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \sum_{n} (\Phi_{\alpha_{n,\varepsilon}}(\gamma))_{\varepsilon} \quad (3.11') $$

is the Fourier-Colombeau transform on $\mathbf{H}$ of the Colombeau measure $(d\mu_{\Phi_{\varepsilon}}(\gamma))_{\varepsilon}$ i.e.

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.12) $$

we can rewrite Eq.(3.11) in the form

$$ (\mathbf{J}[\Phi_{\varepsilon}(\gamma)])_{\varepsilon} = \left( \tilde{\int}_{\mathbf{H}} \exp\left[\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon}. \quad (3.13) $$

We now observe that the right hand side of Eq.(3.13) is well defined also when $\mu_{\Phi_{\varepsilon}}$ is replaced by any bounded for all $\varepsilon \in (0,1]$ Colombeau complex measure on the measurable space $\mathbf{H}$, $\exp[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2]$ being a bounded continuous function. Since any such Colombeau measure $(\mu_{\varepsilon})_{\varepsilon}$ has a Colombeau-Fourier transform on $\mathbf{H}$,
namely a bounded uniformly continuous function $(\Phi_{\mu_{\varepsilon}})_{\varepsilon}$ for all $\varepsilon \in (0,1]$ such that

$$ (\Phi_{\mu_{\varepsilon}}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \qquad (3.14) $$

it is natural to extend the definition of $(\mathbf{J}[\Phi_{\varepsilon}(\gamma)])_{\varepsilon}$ to all such functions $(\Phi_{\varepsilon}(\gamma))_{\varepsilon}$. We then have

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \qquad (3.15) $$

for all functions $\Phi_{\varepsilon}(\gamma)$ on $\mathbf{H}$ which are Colombeau-Fourier transforms of bounded complex Colombeau measures $(\mu_{\varepsilon})_{\varepsilon}$ i.e. such that

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \qquad (3.16) $$

Because of (3.13) It is natural to use for $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ also the notation

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} \qquad (3.17) $$

and to take this, together with Eq.(3.15), as the definition of the normalized Integral (2.1). We have then that the normalized integral is a linear normalized $\tilde{\mathbb{C}}$- valued functional defined at least for all nets $(\Phi_{\varepsilon})_{\varepsilon}$ in the linear vector space $E_M[\mathcal{F}(\mathbf{H})]$ of Colombeau-Fourier transforms of bounded complex Colombeau measures on $\mathbf{H}$, where $\mathcal{F}(\mathbf{H})$ is the linear vector space of Fourier transforms of bounded complex measures on $\mathbf{H}$. The structure of $\mathcal{F}(\mathbf{H})$ is well studied. It is easy to see that $\mathcal{F}(\mathbf{H})$ the linear vector space is a Banach space equipped with the norm $\|\Phi_{\varepsilon}\|_{\mathcal{F}} = \|\mu_{\varepsilon}\|$ for all $\varepsilon \in (0,1]$, where $\|\mu_{\varepsilon}\|$ is the total variation norm for the corresponding measure $\mu_{\varepsilon}$, in fact $\mathcal{F}(\mathbf{H})$ is a Banach function algebra with respect to the
namely a bounded uniformly continuous function $(\Phi_{\mu_{\varepsilon}})_{\varepsilon}$ for all $\varepsilon \in (0,1]$ such that

$$ (\Phi_{\mu_{\varepsilon}}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \quad (3.14) $$

it is natural to extend the definition of $(\mathbf{J}[\Phi_{\varepsilon}(\gamma)])_{\varepsilon}$ to all such functions $(\Phi_{\varepsilon}(\gamma))_{\varepsilon}$. We then have

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\alpha_{\varepsilon}\|^2\right] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \quad (3.15) $$

for all functions $\Phi_{\varepsilon}(\gamma)$ on $\mathbf{H}$ which are Colombeau-Fourier transforms of bounded complex Colombeau measures $(\mu_{\varepsilon})_{\varepsilon}$ i.e. such that

$$ (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle] d\mu_{\varepsilon}(\alpha_{\varepsilon}) \right)_{\varepsilon} \quad (3.16) $$

Because of (3.13) It is natural to use for $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ also the notation

$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} \quad (3.17) $$

and to take this, together with Eq.(3.15), as the definition of the normalized Integral (2.1). We have then that the normalized integral is a linear normalized $\tilde{\mathbb{C}}$- valued functional defined at least for all nets $(\Phi_{\varepsilon})_{\varepsilon}$ in the linear vector space $E_M[\mathcal{F}(\mathbf{H})]$ of Colombeau-Fourier transforms of bounded complex Colombeau measures on $\mathbf{H}$, where $\mathcal{F}(\mathbf{H})$ is the linear vector space of Fourier transforms of bounded complex measures on $\mathbf{H}$. The structure of $\mathcal{F}(\mathbf{H})$ is well studied. It is easy to see that $\mathcal{F}(\mathbf{H})$ the linear vector space is a Banach space equipped with the norm $\|\Phi_{\varepsilon}\|_{\mathcal{F}} = \|\mu_{\varepsilon}\|$ for all $\varepsilon \in (0,1]$, where $\|\mu_{\varepsilon}\|$ is the total variation norm for the corresponding measure $\mu_{\varepsilon}$, in fact $\mathcal{F}(\mathbf{H})$ is a Banach function algebra with respect to the
pointwise multiplication of functions. $\mathcal{F}(\mathbf{H})$ is bijectively isometric with $M(\mathbf{H})$, where $M(\mathbf{H})$ is the space of bounded complex measures on $\mathbf{H}$ equipped with the total variation norm, the bisection being given by the Fourier transform

$\Phi_{\varepsilon}(\gamma) = \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle]d\mu_{\varepsilon}(\alpha_{\varepsilon})$. An Intrinsic characterization of $\mathcal{F}(\mathbf{H})$ is e.g. as the

complex linear hull of those positive definite functions $\Phi_{\varepsilon\in(0,1]}$ which are continuous for all $\varepsilon \in (0,1]$ in the so-called Minlos-Sazonov-Gross topology i.e. those continuous positive definite functions $\Phi_{\varepsilon\in(0,1]}$ on $\mathbf{H}$ for which for any $\delta > 0$ there exists a nuclear operator (positive (symmetric) trace class) $N_{\delta}$ such that $\mathrm{Re}(\Phi_{\varepsilon}(0) - \Phi_{\varepsilon}(\gamma)) < \delta$ whenever $\langle\gamma, N_{\delta}\gamma\rangle < 1$. We call $\mathcal{F}(\mathbf{H})$ the space of (Fresnel) integrable functions [38],[39] and we call $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ the (Colombeau-Fresnel) integral of $(\Phi_{\varepsilon})_{\varepsilon}$. $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ is the normalized integral of $(\Phi_{\varepsilon}(\cdot)\exp[\frac{i}{2}\|\cdot\|^2])_{\varepsilon}$ illustrated above. The properties of this Colombeau integral follows directly from the properties of the Fresnel integral which are studied in [39].

**▲ P. 1.** $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ is a $\tilde{\mathbb{C}}$- linear normalized bounded functional on the Colombeau-Banach module $G_{\mathcal{F}(\mathbf{H})}$. Thus in particular we have

$$ \left| \left( \mathbf{J} \left[ \prod_{j=1}^{n} \Phi_{j, \varepsilon} \right] \right)_{\varepsilon \in (0,1]} \right| \le \prod_{j=1}^{n} (\| \Phi_{j, \varepsilon} \|_{\mathcal{F}})_{\varepsilon \in (0,1]} \quad (3.18) $$

where $|\cdot|$ on the left hand side is the absolute value in the field $\mathbb{C}$ of complex numbers and we recall that $\|\cdot\|$ is the norm in $\mathcal{F}(\mathbf{H})$.

**▲ P.2.** When $\mathbf{H}$ is finite dimensional i.e. $\mathbf{H} = \mathbb{R}^d$ then

$$ \begin{aligned} (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} &= \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} = \\ &\qquad (2\pi i)^{-d/2} \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) d\gamma \right)_{\varepsilon} \end{aligned} \quad (3.19) $$

for all $(\Phi_{\varepsilon})_{\varepsilon}$ in $E_M[\mathcal{F}(\mathbb{R}^d)]$ for which the Colombeau-Lebesgue integral exists. Another useful property is the linearity of $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ under smooth partitions $\{\varphi_k\}$ of the unit i.e.
pointwise multiplication of functions. $\mathcal{F}(\mathbf{H})$ is bijectively isometric with $M(\mathbf{H})$, where $M(\mathbf{H})$ is the space of bounded complex measures on $\mathbf{H}$ equipped with the total variation norm, the bisection being given by the Fourier transform

$\Phi_{\varepsilon}(\gamma) = \int_{\mathbf{H}} \exp[i\langle\gamma, \alpha_{\varepsilon}\rangle]d\mu_{\varepsilon}(\alpha_{\varepsilon})$. An Intrinsic characterization of $\mathcal{F}(\mathbf{H})$ is e.g. as the

complex linear hull of those positive definite functions $\Phi_{\varepsilon \in (0,1]}$ which are continuous for all $\varepsilon \in (0,1]$ in the so-called Minlos-Sazonov-Gross topology i.e. those continuous positive definite functions $\Phi_{\varepsilon \in (0,1]}$ on $\mathbf{H}$ for which for any $\delta > 0$ there exists a nuclear operator (positive (symmetric) trace class) $N_{\delta}$ such that $\mathrm{Re}(\Phi_{\varepsilon}(0) - \Phi_{\varepsilon}(\gamma)) < \delta$ whenever $\langle\gamma, N_{\delta}\gamma\rangle < 1$. We call $\mathcal{F}(\mathbf{H})$ the space of (Fresnel) integrable functions [38],[39] and we call $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ the (Colombeau-Fresnel) integral of $(\Phi_{\varepsilon})_{\varepsilon}$. $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ is the normalized integral of $(\Phi_{\varepsilon}(\cdot)\exp[\frac{i}{2}\|\cdot\|^2])_{\varepsilon}$ illustrated above. The properties of this Colombeau integral follows directly from the properties of the Fresnel integral which are studied in [39].

**▲ P. 1.** $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ is a $\tilde{\mathbb{C}}$- linear normalized bounded functional on the Colombeau-Banach module $G_{\mathcal{F}(\mathbf{H})}$. Thus in particular we have

$$ \left| \left( \mathbf{J} \left[ \prod_{j=1}^{n} \Phi_{j, \varepsilon} \right] \right)_{\varepsilon \in (0,1]} \right| \leq \prod_{j=1}^{n} (\| \Phi_{j, \varepsilon} \|_{\mathcal{F}})_{\varepsilon \in (0,1]} \qquad (3.18) $$

where $|\cdot|$ on the left hand side is the absolute value in the field $\mathbb{C}$ of complex numbers and we recall that $\|\cdot\|$ is the norm in $\mathcal{F}(\mathbf{H})$.

**▲ P.2.** When $\mathbf{H}$ is finite dimensional i.e. $\mathbf{H} = \mathbb{R}^d$ then

$$ \begin{aligned} (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} &= \left( \int_{\mathbf{H}} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) D[\gamma] \right)_{\varepsilon} = \\ &\qquad (2\pi i)^{-d/2} \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) d\gamma \right)_{\varepsilon} \end{aligned} \qquad (3.19) $$

for all $(\Phi_{\varepsilon})_{\varepsilon}$ in $E_M[\mathcal{F}(\mathbb{R}^d)]$ for which the Colombeau-Lebesgue integral exists. Another useful property is the linearity of $(\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon}$ under smooth partitions $\{\varphi_k\}$ of the unit i.e.
$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) d\gamma \right)_{\varepsilon} = \\ \sum_k \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) \varphi_k(\gamma) d\gamma \right)_{\varepsilon} = \qquad (3.20) \\ \sum_k (\mathbf{J}[\Phi_{\varepsilon} \varphi_k])_{\varepsilon} $$

with $\varphi_k(\gamma)$ of compact support (e.g. $\varphi_k(\gamma) = \psi_k(\gamma) - \psi_{k-1}(\gamma)$, $k \in \mathbb{N}$, $\psi_0(\gamma) = 0$, $\psi_k(\gamma) = \psi(\gamma/k)$, $\psi(\gamma)$ being a $C^{\infty}$ function equal to 0 for $\|\gamma\| > 2$ and equal to 1 for $\|\gamma\| < 1$.

**▲ P.3.** Let $\mathcal{F}_P(\mathbf{H})$ be the subalgebra of $\mathcal{F}(\mathbf{H})$ consisting of functions $f$ in $\mathcal{F}(\mathbf{H})$ with the property that $f(\gamma) = f(P_f\gamma)$ for some individual finite dimensional (i.e. with finite rank=n) projection on $\mathbf{H}$. i.e. the functions on $\mathcal{F}_P(\mathbf{H})$ are the cylinder or tame functions on $\mathbf{H}$. Let for $(\Phi_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$ be $(\mathbf{J}_{P_{\Phi_{\varepsilon}}\mathbf{H}}[\cdot])_{\varepsilon}$ the normalized integral on the finite dimensional space $E_M(P_{\Phi_{\varepsilon}}\mathbf{H})$. Then $(\mathbf{J}_{\mathbf{H}}[\Phi_{\varepsilon}])_{\varepsilon} = (\mathbf{J}_{P_{\Phi_{\varepsilon}}\mathbf{H}}[\tilde{\Phi}_{\varepsilon}])_{\varepsilon}$, where $\tilde{\Phi}_{\varepsilon}, \varepsilon \in (0,1]$ is the function on $P_{\Phi_{\varepsilon}}\mathbf{H}$ such that $(\tilde{\Phi}_{\varepsilon}(P_{\Phi_{\varepsilon}}\gamma))_{\varepsilon} = ((\Phi_{\varepsilon} \circ P_{\Phi_{\varepsilon}})(\gamma))_{\varepsilon}$. In particular for $(\Phi_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$

$$ (\mathbf{J}_{\mathbf{H}}[\Phi_{\varepsilon}])_{\varepsilon} = (2\pi i)^{-(\dim P_{\Phi_{\varepsilon}}\mathbf{H})} \int_{\mathbb{R}^d} \tilde{\Phi}_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \|\gamma\|^2\right] d\gamma. \qquad (3.21) $$

whenever the Riemann or Lebesgue integral on the right hand side exists.

**▲ P.4.** The translation invariance of the normalized integral has been built in from the beginning in our presentation above. One has thus

$$ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma + \alpha_{\varepsilon}) \exp\left[ \frac{i}{2} \|\gamma + \alpha_{\varepsilon}\|^2 \right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} \qquad (3.22) $$

for all $(\alpha_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$. Moreover for any bounded operator $\mathbf{A} : \mathbf{H} \to \mathbf{H}$ with everywhere define bounded inverse we have
$$ (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) d\gamma \right)_{\varepsilon} = \\ \sum_k \left( \int_{\mathbb{R}^d} \exp\left[-\frac{i}{2} \|\gamma\|^2\right] \Phi_{\varepsilon}(\gamma) \varphi_k(\gamma) d\gamma \right)_{\varepsilon} = \qquad (3.20) \\ \sum_k (\mathbf{J}[\Phi_{\varepsilon} \varphi_k])_{\varepsilon} $$

with $\varphi_k(\gamma)$ of compact support (e.g. $\varphi_k(\gamma) = \psi_k(\gamma) - \psi_{k-1}(\gamma)$, $k \in \mathbb{N}$, $\psi_0(\gamma) = 0$, $\psi_k(\gamma) = \psi(\gamma/k)$, $\psi(\gamma)$ being a $C^{\infty}$ function equal to 0 for $\|\gamma\| > 2$ and equal to 1 for $\|\gamma\| < 1$.

**▲ P.3.** Let $\mathcal{F}_P(\mathbf{H})$ be the subalgebra of $\mathcal{F}(\mathbf{H})$ consisting of functions $f$ in $\mathcal{F}(\mathbf{H})$ with the property that $f(\gamma) = f(P_f\gamma)$ for some individual finite dimensional (i.e. with finite rank=n) projection on $\mathbf{H}$. i.e. the functions on $\mathcal{F}_P(\mathbf{H})$ are the cylinder or tame functions on $\mathbf{H}$. Let for $(\Phi_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$ be $(\mathbf{J}_{P_{\Phi_{\varepsilon}}\mathbf{H}}[\cdot])_{\varepsilon}$ the normalized integral on the finite dimensional space $E_M(P_{\Phi_{\varepsilon}}\mathbf{H})$. Then $(\mathbf{J}_{\mathbf{H}}[\Phi_{\varepsilon}])_{\varepsilon} = (\mathbf{J}_{P_{\Phi_{\varepsilon}}\mathbf{H}}[\tilde{\Phi}_{\varepsilon}])_{\varepsilon}$, where $\tilde{\Phi}_{\varepsilon}, \varepsilon \in (0,1]$ is the function on $P_{\Phi_{\varepsilon}}\mathbf{H}$ such that $(\tilde{\Phi}_{\varepsilon}(P_{\Phi_{\varepsilon}}\gamma))_{\varepsilon} = ((\Phi_{\varepsilon} \circ P_{\Phi_{\varepsilon}})(\gamma))_{\varepsilon}$. In particular for $(\Phi_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$

$$ (\mathbf{J}_{\mathbf{H}}[\Phi_{\varepsilon}])_{\varepsilon} = (2\pi i)^{-(\dim P_{\Phi_{\varepsilon}}\mathbf{H})} \int_{\mathbb{R}^d} \tilde{\Phi}_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \|\gamma\|^2\right] d\gamma. \qquad (3.21) $$

whenever the Riemann or Lebesgue integral on the right hand side exists.

**▲ P.4.** The translation invariance of the normalized integral has been built in from the beginning in our presentation above. One has thus

$$ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma + \alpha_{\varepsilon}) \exp\left[ \frac{i}{2} \|\gamma + \alpha_{\varepsilon}\|^2 \right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} \qquad (3.22) $$

for all $(\alpha_{\varepsilon})_{\varepsilon} \in E_M(\mathcal{F}_P(\mathbf{H}))$. Moreover for any bounded operator $\mathbf{A}: \mathbf{H} \to \mathbf{H}$ with everywhere define bounded inverse we have
$$ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\mathbf{A}\gamma) \exp\left[ \frac{i}{2} \|\mathbf{A}\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} \quad (3.23) $$

In particular one has the Euclidean invariance of the normalized integral.

**▲ P. 5.** For the normalized Colombeau integral a "Fubini Theorem" concerning iterated integration

$$ \left( \int_{H_1 \times H_2} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[\frac{i}{2} \|\gamma\|^2\right] D[\gamma_1] D[\gamma_2] \right)_{\varepsilon} = \left( \int_{H_2} \exp\left[\frac{i}{2} \|\gamma_2\|^2\right] \left\{ \int_{H_1} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[\frac{i}{2} \|\gamma_1\|^2\right] D[\gamma_1] \right\} D[\gamma_2] \right)_{\varepsilon} = \quad (3.24) $$

$$ \left( \int_{H_2} \exp\left[\frac{i}{2} \|\gamma_1\|^2\right] \left\{ \int_{H_1} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[\frac{i}{2} \|\gamma_2\|^2\right] D[\gamma_2] \right\} D[\gamma_1] \right)_{\varepsilon}. $$

## The nofinalized Feynman-Colombeau path integral with respect to a quadratic form.

In order to treat on the same footing quantum mechanical systems in which a quadratic term in the coordinates arises in the action (and in the Hamiltonian) it is useful to introduce the concept of an Feynman-Colombeau Integral normalized with respect to a (non necessarily positive) quadratic form.

Let us think as an example to a system in $\mathbb{R}^d$ with classical action $(S_T(\gamma))_{\varepsilon}, \varepsilon \in (0, 1]$ for the time Interval $[0, t]$
$$ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\mathbf{A}\gamma) \exp\left[ \frac{i}{2} \|\mathbf{A}\gamma\|^2 \right] D[\gamma] \right)_{\varepsilon} \quad (3.23) $$

In particular one has the Euclidean invariance of the normalized integral.

**▲ P. 5.** For the normalized Colombeau integral a "Fubini Theorem" concerning iterated integration

$$ \left( \int_{H_1 \times H_2} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[ \frac{i}{2} \|\gamma\|^2 \right] D[\gamma_1] D[\gamma_2] \right)_{\varepsilon} = $$

$$ \left( \int_{H_2} \exp\left[ \frac{i}{2} \|\gamma_2\|^2 \right] \left\{ \int_{H_1} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[ \frac{i}{2} \|\gamma_1\|^2 \right] D[\gamma_1] \right\} D[\gamma_2] \right)_{\varepsilon} = \quad (3.24) $$

$$ \left( \int_{H_2} \exp\left[ \frac{i}{2} \|\gamma_1\|^2 \right] \left\{ \int_{H_1} \Phi_{\varepsilon}(\gamma_1, \gamma_2) \exp\left[ \frac{i}{2} \|\gamma_2\|^2 \right] D[\gamma_2] \right\} D[\gamma_1] \right)_{\varepsilon}. $$

## The nofinalized Feynman-Colombeau path integral with respect to a quadratic form.

In order to treat on the same footing quantum mechanical systems in which a quadratic term in the coordinates arises in the action (and in the Hamiltonian) it is useful to introduce the concept of an Feynman-Colombeau Integral normalized with respect to a (non necessarily positive) quadratic form.

Let us think as an example to a system in $\mathbb{R}^d$ with classical action $(S_T(\gamma))_{\varepsilon}, \varepsilon \in (0,1]$ for the time Interval $[0,t]$
$$ (S_{T,\varepsilon}(\gamma))_{\varepsilon} = (S_{T,\varepsilon}^{0}(\gamma))_{\varepsilon} + \left( \int_{0}^{T} V_{\varepsilon}(\gamma(t))dt \right)_{\varepsilon}, \qquad (3.25) $$

where

$$ (S_{T,\varepsilon}^{0}(\gamma))_{\varepsilon} = \frac{1}{2} \int_{0}^{T} \dot{\gamma}^{2}(t)dt - \frac{1}{2} \left( \int_{0}^{T} (\gamma(t), A_{\varepsilon}\gamma(t))dt \right)_{\varepsilon} \qquad (3.26) $$

Here $A_{\varepsilon}$ is a $d \times d$ matrix. Since quadratic functions are not Fourier transform of bounded complex measures, it is natural to let the quadratic form $S_{T,\varepsilon}^{0}(\gamma)$ take the role of the kinetic energy term in the previous definition of normalized integral (Feynman path integrals for the case under consideration). Mathematically it turns out that this can be done as a particular case of the following situation. Let $\mathbf{H}$ be a real separable Hilbert space. Let $(B_{\varepsilon})_{\varepsilon}$ be an net a densely defined symmetric operators in $\mathbf{H}$, then $\langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}$ is a real quadratic form on $\mathbf{D}(B_{\varepsilon})$. He would like to define, somewhat in analogy with above construction a normalized Colombeau integral

$$ \left( \tilde{\int}_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp \left[ \frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}} \right] D[\gamma] \right)_{\varepsilon}, \qquad (3.27) $$

where $\langle \cdot, \cdot \rangle_{\mathbf{H}}$ is the scalar product in $\mathbf{H}$. The normalization should be such that

$$ \mathrm{cl} \left[ \left( \tilde{\int}_{\mathbf{H}} \exp \left[ \frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}} \right] D[\gamma] \right)_{\varepsilon} \right] = 1. \qquad (3.28) $$

From the care $\mathbf{H} = \mathbb{R}^d$ see that it is natural to assume that all $B_{\varepsilon}$, $\varepsilon \in (0,1]$ it non degenerate, namely in order to define (3.27) In analogy with Eq.(3.13),i.e.by a Parseval relation involving the Colombeau-Fourier transfonws of $\exp[-\frac{i}{2}\langle \gamma, B_{\varepsilon}\gamma \rangle]$ Let us

look first at the situation where $B_{\varepsilon}$ is strictly positive and bounded everywhere in $\mathbf{H}$. Then, with $\langle \gamma, \gamma \rangle_{B_{\varepsilon}} = \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}$ the closure of $\mathbf{H}$ in the $\|\cdot\|_{B_{\varepsilon}}$-norm, we have by Eq.(3.13)
$$ (S_{T,\varepsilon}(\gamma))_{\varepsilon} = (S_{T,\varepsilon}^{0}(\gamma))_{\varepsilon} + \left( \int_{0}^{T} V_{\varepsilon}(\gamma(t))dt \right)_{\varepsilon}, \qquad (3.25) $$

where

$$ (S_{T,\varepsilon}^{0}(\gamma))_{\varepsilon} = \frac{1}{2} \int_{0}^{T} \dot{\gamma}^{2}(t)dt - \frac{1}{2} \left( \int_{0}^{T} (\gamma(t), A_{\varepsilon}\gamma(t))dt \right)_{\varepsilon} \qquad (3.26) $$

Here $A_{\varepsilon}$ is a $d \times d$ matrix. Since quadratic functions are not Fourier transform of bounded complex measures, it is natural to let the quadratic form $S_{T,\varepsilon}^{0}(\gamma)$ take the role of the kinetic energy term in the previous definition of normalized integral (Feynman path integrals for the case under consideration). Mathematically it turns out that this can be done as a particular case of the following situation. Let $\mathbf{H}$ be a real separable Hilbert space. Let $(B_{\varepsilon})_{\varepsilon}$ be an net a densely defined symmetric operators in $\mathbf{H}$, then $\langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}$ is a real quadratic form on $\mathbf{D}(B_{\varepsilon})$. He would like to define, somewhat in analogy with above construction a normalized Colombeau integral

$$ \left( \tilde{\int}_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp \left[ \frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}} \right] D[\gamma] \right)_{\varepsilon}, \qquad (3.27) $$

where $\langle \cdot, \cdot \rangle_{\mathbf{H}}$ is the scalar product in $\mathbf{H}$. The normalization should be such that

$$ \mathrm{cl} \left[ \left( \tilde{\int}_{\mathbf{H}} \exp \left[ \frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}} \right] D[\gamma] \right)_{\varepsilon} \right] = 1. \qquad (3.28) $$

From the care $\mathbf{H} = \mathbb{R}^d$ see that it is natural to assume that all $B_{\varepsilon}$, $\varepsilon \in (0,1]$ it non degenerate, namely in order to define (3.27) In analogy with Eq.(3.13),i.e.by a Parseval relation involving the Colombeau-Fourier transfonws of $\exp[-\frac{i}{2}\langle \gamma, B_{\varepsilon}\gamma \rangle]$ Let us

look first at the situation where $B_{\varepsilon}$ is strictly positive and bounded everywhere in $\mathbf{H}$. Then, with $\langle \gamma, \gamma \rangle_{B_{\varepsilon}} = \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}$ the closure of $\mathbf{H}$ in the $\|\cdot\|_{B_{\varepsilon}}$-norm, we have by Eq.(3.13)
$$ \left( \tilde{\int}_{H_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \langle \gamma, \gamma \rangle_{B_{\varepsilon}} \right] d\gamma \right)_{\varepsilon} = \left( \int_{H_{B_{\varepsilon}}} \Phi_{\varepsilon}(\alpha) \exp\left[ -\frac{i}{2} \langle \alpha, \alpha \rangle_{B_{\varepsilon}} \right] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon} \quad (3.29) $$

for any $\Phi_{\varepsilon}(\gamma)$ such that $\Phi_{\varepsilon}(\gamma) \in \mathcal{F}(H_{B_{\varepsilon}})$, $\varepsilon \in (0,1]$ and where $\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon})$ is the Colombeau measure whose Colombeau-Fourier transform is $(\Phi_{\varepsilon}(\gamma))_{\varepsilon}$ i.e.

$$ \begin{gathered} (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{H} \exp[i\langle\gamma, \alpha\rangle_{B_{\varepsilon}}] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon} = \\ \left( \int_{H} \exp[i\langle\gamma, \alpha\rangle_{H}] d\mu_{\Phi_{\varepsilon}}(\alpha) \right)_{\varepsilon}, \end{gathered} \quad (3.30) $$

where

$$ d\mu_{\Phi_{\varepsilon}}(\alpha) = d\mu_{\Phi_{\varepsilon}}(B_{\varepsilon}^{-1}\alpha, B_{\varepsilon}). \quad (3.31) $$

From (3.29) - (3.31) we get
$$ \left( \tilde{\int}_{H_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[ \frac{i}{2} \langle \gamma, \gamma \rangle_{B_{\varepsilon}} \right] d\gamma \right)_{\varepsilon} = \left( \int_{H_{B_{\varepsilon}}} \Phi_{\varepsilon}(\alpha) \exp\left[ -\frac{i}{2} \langle \alpha, \alpha \rangle_{B_{\varepsilon}} \right] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon} \quad (3.29) $$

for any $\Phi_{\varepsilon}(\gamma)$ such that $\Phi_{\varepsilon}(\gamma) \in \mathcal{F}(H_{B_{\varepsilon}})$, $\varepsilon \in (0,1]$ and where $\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon})$ is the Colombeau measure whose Colombeau-Fourier transform is $(\Phi_{\varepsilon}(\gamma))_{\varepsilon}$ i.e.

$$ \begin{gathered} (\Phi_{\varepsilon}(\gamma))_{\varepsilon} = \left( \int_{H} \exp[i\langle\gamma, \alpha\rangle_{B_{\varepsilon}}] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon} = \\ \left( \int_{H} \exp[i\langle\gamma, \alpha\rangle_{H}] d\mu_{\Phi_{\varepsilon}}(\alpha) \right)_{\varepsilon}, \end{gathered} \quad (3.30) $$

where

$$ d\mu_{\Phi_{\varepsilon}}(\alpha) = d\mu_{\Phi_{\varepsilon}}(B_{\varepsilon}^{-1}\alpha, B_{\varepsilon}). \quad (3.31) $$

From (3.29) - (3.31) we get
$$
\begin{align*}
& \left( \left( \int_{H_{B_\varepsilon}} \Phi_\varepsilon(\gamma) \exp\left[ \frac{i}{2} \langle \gamma, \gamma \rangle_{B_\varepsilon} \right] d\gamma \right)_\varepsilon \right) = \\
& \left( \left( \int_{H_{B_\varepsilon}} \Phi_\varepsilon(\alpha) \exp\left[ -\frac{i}{2} \langle \alpha, B_\varepsilon \alpha \rangle_H \right] d\mu_{\Phi_\varepsilon}(\alpha, B_\varepsilon) \right)_\varepsilon \right) = \\
& \left( \left( \int_{R(B_\varepsilon)} \Phi_\varepsilon(\beta) \exp\left[ -\frac{i}{2} \langle B_\varepsilon^{-1} \beta, \beta \rangle_H \right] d\mu_{\Phi_\varepsilon}(\beta) \right)_\varepsilon \right) = \\
& \left( \left( \int_{R(B_\varepsilon)} \Phi_\varepsilon(\beta) \exp\left[ -\frac{i}{2} \Delta_\varepsilon(\beta, \beta) \right] d\mu_{\Phi_\varepsilon}(\beta) \right)_\varepsilon \right)
\end{align*}
\tag{3.32}
$$

with $\Delta_\varepsilon(\beta, \beta)$ the bilinear form such that $\Delta_\varepsilon(\beta, \beta) = \langle B_\varepsilon^{-1} \beta, \beta \rangle_H$ and $R(B_\varepsilon)$ means the range of $B_\varepsilon$. Hence the normalized integral on $H_{B_\varepsilon}$ is given by a "Parseval relation" involving an integral of $\exp[-\frac{i}{2}\Delta_\varepsilon(\beta, \beta)]$ against a bounded complex measure. This suggests to define in general (3.27) by

$$ \left( \left( \int_{D_{\varepsilon}} \Phi_{\varepsilon}(\beta) \exp\left[-\frac{i}{2} \Delta_{\varepsilon}(\beta, \beta)\right] d\mu_{\Phi_{\varepsilon}}(\beta) \right) \right)_{\varepsilon} \qquad (3.33) $$

where $D_\varepsilon$ is a suitable linear subset of $H$ and $\Delta_\varepsilon$ is a suitable "inverse" of $B_\varepsilon$. By the above considerations we see, that it is natural to require that $\Delta_\varepsilon$ be a symmetric bilinear form on $D_\varepsilon \times D_\varepsilon$, such that $\Delta_\varepsilon(\alpha, B_\varepsilon \beta) = \langle \alpha, \beta \rangle_H$,

$\forall \alpha \in H, \beta \in D_\varepsilon = D(B_\varepsilon)$, assuming that the range of $B_\varepsilon$ is contained in $D_\varepsilon$. For applications to the case of QM measurments it is useful to allow $(\Delta_\varepsilon)_\varepsilon$ to be a $\tilde{C}$-valued and we see that the simplest assumption to define (3.33) is to require that the integration be taken on some nice space, e.g. $D_\varepsilon$ a separable Banach space densely contained in $H$ and with norm $\|\cdot\|_{D_\varepsilon}$ stronger than the one $\|\cdot\|_H$, in $H$ (i.e. $\|\cdot\|_H < \delta_\varepsilon \|\cdot\|_{D_\varepsilon}$ for some $\delta_\varepsilon$), and with $\Delta_\varepsilon$ continuous on $D_\varepsilon \times D_\varepsilon$, with $\mathrm{Im}\,\Delta_\varepsilon(\gamma, \gamma) \le 0$. The definition of (3.27) by (3.33) clearly depends on the choice of $\Delta_\varepsilon$, hence it is useful to write
$$
\begin{align*}
& \left( \left( \int_{H_{B_\varepsilon}} \Phi_\varepsilon(\gamma) \exp\left[ \frac{i}{2} \langle \gamma, \gamma \rangle_{B_\varepsilon} \right] d\gamma \right)_\varepsilon \right) = \\
& \left( \left( \int_{H_{B_\varepsilon}} \Phi_\varepsilon(\alpha) \exp\left[ -\frac{i}{2} \langle \alpha, B_\varepsilon \alpha \rangle_H \right] d\mu_{\Phi_\varepsilon}(\alpha, B_\varepsilon) \right)_\varepsilon \right) = \\
& \left( \left( \int_{R(B_\varepsilon)} \Phi_\varepsilon(\beta) \exp\left[ -\frac{i}{2} \langle B_\varepsilon^{-1} \beta, \beta \rangle_H \right] d\mu_{\Phi_\varepsilon}(\beta) \right)_\varepsilon \right) = \\
& \left( \left( \int_{R(B_\varepsilon)} \Phi_\varepsilon(\beta) \exp\left[ -\frac{i}{2} \Delta_\varepsilon(\beta, \beta) \right] d\mu_{\Phi_\varepsilon}(\beta) \right)_\varepsilon \right)
\end{align*}
\tag{3.32}
$$

with $\Delta_\varepsilon(\beta, \beta)$ the bilinear form such that $\Delta_\varepsilon(\beta, \beta) = \langle B_\varepsilon^{-1} \beta, \beta \rangle_H$ and $R(B_\varepsilon)$ means the range of $B_\varepsilon$. Hence the normalized integral on $H_{B_\varepsilon}$ is given by a "Parseval relation" involving an integral of $\exp[-\frac{i}{2}\Delta_\varepsilon(\beta, \beta)]$ against a bounded complex measure. This suggests to define in general (3.27) by

$$ \left( \left( \int_{D_{\varepsilon}} \Phi_{\varepsilon}(\beta) \exp\left[-\frac{i}{2} \Delta_{\varepsilon}(\beta, \beta)\right] d\mu_{\Phi_{\varepsilon}}(\beta) \right) \right)_{\varepsilon} \qquad (3.33) $$

where $D_\varepsilon$ is a suitable linear subset of $H$ and $\Delta_\varepsilon$ is a suitable "inverse" of $B_\varepsilon$. By the above considerations we see, that it is natural to require that $\Delta_\varepsilon$ be a symmetric bilinear form on $D_\varepsilon \times D_\varepsilon$, such that $\Delta_\varepsilon(\alpha, B_\varepsilon \beta) = \langle \alpha, \beta \rangle_H$,

$\forall \alpha \in H, \beta \in D_\varepsilon = D(B_\varepsilon)$, assuming that the range of $B_\varepsilon$ is contained in $D_\varepsilon$. For applications to the case of QM measurements it is useful to allow $(\Delta_\varepsilon)_\varepsilon$ to be a $\tilde{C}$-valued and we see that the simplest assumption to define (3.33) is to require that the integration be taken on some nice space, e.g. $D_\varepsilon$ a separable Banach space densely contained in $H$ and with norm $\|\cdot\|_{D_\varepsilon}$ stronger than the one $\|\cdot\|_H$, in $H$ (i.e. $\|\cdot\|_H < \delta_\varepsilon \|\cdot\|_{D_\varepsilon}$ for some $\delta_\varepsilon$), and with $\Delta_\varepsilon$ continuous on $D_\varepsilon \times D_\varepsilon$, with $\mathrm{Im}\,\Delta_\varepsilon(\gamma, \gamma) \le 0$. The definition of (3.27) by (3.33) clearly depends on the choice of $\Delta_\varepsilon$, hence it is useful to write
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{D}_{\varepsilon}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\Delta_{\varepsilon}(\gamma, B_{\varepsilon}\gamma)\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.34) $$

instead of (3.27). Hence we come to the definition:

**Definition.** Let $(\mathbf{H}, (\mathbf{D}_{\varepsilon})_{\varepsilon}, (\|\cdot\|_{\varepsilon})_{\varepsilon}, (B_{\varepsilon})_{\varepsilon}, \Delta_{\varepsilon}, \varepsilon \in (0, 1]$, be a "Fresnel-Colombeau fourtuple" as described above, i.e. $\mathbf{H}$ a real separable Hilbert space, $\mathbf{D}_{\varepsilon}$ a real separable Banach space densely contained in $\mathbf{H}$ for all $\varepsilon \in (0, 1]$ and with norm $\|\cdot\|_{\varepsilon}$ dominating the one in $\mathbf{H}, (B_{\varepsilon})_{\varepsilon \in (0,1]}$ a net of a symmetric densely defined operators (not necessarily bounded) with range containing $\mathbf{D}_{\varepsilon}$ and $(\Delta_{\varepsilon})_{\varepsilon}$ a $\tilde{\mathbb{C}}$-valued symmetric bilinear form with non positive imaginary part and satisfying $\Delta_{\varepsilon}(\alpha, B_{\varepsilon}\beta) = \langle\alpha, \beta\rangle_{\mathbf{H}}$. Let $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$ be the Banach algebra of functions $\Phi_{\varepsilon}$ on the topological dual $\mathbf{D}_{\varepsilon}^*$ of $\mathbf{D}_{\varepsilon}$ which are Fourier transforms of bounded complex measures $\mu_{\Phi_{\varepsilon}}$ on $\mathbf{D}_{\varepsilon}$ i.e. $\Phi_{\varepsilon}(\gamma) = \int \exp[i\langle\gamma, \alpha\rangle]d\mu_{\Phi_{\varepsilon}}(\alpha)$, where $\langle\cdot, \cdot\rangle$ is the dualization between $\mathbf{D}_{\varepsilon}$ and $\mathbf{D}_{\varepsilon}^*$. Multiplication of functions is, as in $\mathcal{F}(\mathbf{H})$, the pointwise one, and the norm $\|\cdot\|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)}$ in $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$ is defined as the total variation $\|\mu_{\Phi_{\varepsilon}}\|$ in the corresponding Banach algebra $M(\mathbf{D}_{\varepsilon}^*)$ of bounded complex measures $\mu_{\Phi_{\varepsilon}}$ on $\mathbf{D}_{\varepsilon}^*$. We call $(\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon}$ as normalized Colombeau integral with respect to the quadratic form $(\Delta_{\varepsilon})_{\varepsilon}$. This integral enjoys properties related to those of the normalized integral defined by (3.17).

**▲ P.1.** $\Phi_{\varepsilon} \to \mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}]$ for all $\varepsilon \in (0, 1]$ is a $\tilde{\mathbb{C}}$-valued, linear, bounded, continuous normalized functional on the Banach function algebra $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$. One has

$$ \left| \left( \mathbf{J}_{\Delta_{\varepsilon}} \left[ \prod_{j=1}^{n} \Phi_{j, \varepsilon} \right] \right)_{\varepsilon \in (0,1]} \right| \le \prod_{j=1}^{n} \left( \| \Phi_{j, \varepsilon} \|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)} \right)_{\varepsilon \in (0,1]} \quad (3.35) $$

with $\|\cdot\|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)} = \|\mu_{\Phi_{\varepsilon}}\| = \text{total variation on } \mathbf{D}_{\varepsilon}$ of $\mu_{\Phi_{\varepsilon}}$.

**▲ P.2.** When $B_{\varepsilon}$ is strictly positive and with range dense in $\mathbf{D}_{\varepsilon}$ for all $\varepsilon \in (0, 1]$ then

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle_{B_{\varepsilon}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.36) $$

In particular for $\mathfrak{cl}[(B_{\varepsilon})_{\varepsilon}] = 1$
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{D}_{\varepsilon}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\Delta_{\varepsilon}(\gamma, B_{\varepsilon}\gamma)\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.34) $$

instead of (3.27). Hence we come to the definition:

**Definition.** Let $(\mathbf{H}, (\mathbf{D}_{\varepsilon})_{\varepsilon}, (\|\cdot\|_{\varepsilon})_{\varepsilon}, (B_{\varepsilon})_{\varepsilon}, \Delta_{\varepsilon}, \varepsilon \in (0, 1]$, be a "Fresnel-Colombeau fourtuple" as described above, i.e. $\mathbf{H}$ a real separable Hilbert space, $\mathbf{D}_{\varepsilon}$ a real separable Banach space densely contained in $\mathbf{H}$ for all $\varepsilon \in (0, 1]$ and with norm $\|\cdot\|_{\varepsilon}$ dominating the one in $\mathbf{H}, (B_{\varepsilon})_{\varepsilon \in (0,1]}$ a net of a symmetric densely defined operators (not necessarily bounded) with range containing $\mathbf{D}_{\varepsilon}$ and $(\Delta_{\varepsilon})_{\varepsilon}$ a $\tilde{\mathbb{C}}$-valued symmetric bilinear form with non positive imaginary part and satisfying $\Delta_{\varepsilon}(\alpha, B_{\varepsilon}\beta) = \langle\alpha, \beta\rangle_{\mathbf{H}}$. Let $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$ be the Banach algebra of functions $\Phi_{\varepsilon}$ on the topological dual $\mathbf{D}_{\varepsilon}^*$ of $\mathbf{D}_{\varepsilon}$ which are Fourier transforms of bounded complex measures $\mu_{\Phi_{\varepsilon}}$ on $\mathbf{D}_{\varepsilon}$ i.e. $\Phi_{\varepsilon}(\gamma) = \int \exp[i\langle\gamma, \alpha\rangle]d\mu_{\Phi_{\varepsilon}}(\alpha)$, where $\langle\cdot, \cdot\rangle$ is the dualization between $\mathbf{D}_{\varepsilon}$ and $\mathbf{D}_{\varepsilon}^*$. Multiplication of functions is, as in $\mathcal{F}(\mathbf{H})$, the pointwise one, and the norm $\|\cdot\|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)}$ in $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$ is defined as the total variation $\|\mu_{\Phi_{\varepsilon}}\|$ in the corresponding Banach algebra $M(\mathbf{D}_{\varepsilon}^*)$ of bounded complex measures $\mu_{\Phi_{\varepsilon}}$ on $\mathbf{D}_{\varepsilon}^*$. We call $(\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon}$ as normalized Colombeau integral with respect to the quadratic form $(\Delta_{\varepsilon})_{\varepsilon}$. This integral enjoys properties related to those of the normalized integral defined by (3.17).

**▲ P.1.** $\Phi_{\varepsilon} \to \mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}]$ for all $\varepsilon \in (0, 1]$ is a $\tilde{\mathbb{C}}$-valued, linear, bounded, continuous normalized functional on the Banach function algebra $\mathcal{F}(\mathbf{D}_{\varepsilon}^*)$. One has

$$ \left| \left( \mathbf{J}_{\Delta_{\varepsilon}} \left[ \prod_{j=1}^{n} \Phi_{j, \varepsilon} \right] \right)_{\varepsilon \in (0,1]} \right| \le \prod_{j=1}^{n} \left( \| \Phi_{j, \varepsilon} \|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)} \right)_{\varepsilon \in (0,1]} \quad (3.35) $$

with $\|\cdot\|_{\mathcal{F}(\mathbf{D}_{\varepsilon}^*)} = \|\mu_{\Phi_{\varepsilon}}\| = \text{total variation on } \mathbf{D}_{\varepsilon}$ of $\mu_{\Phi_{\varepsilon}}$.

**▲ P.2.** When $B_{\varepsilon}$ is strictly positive and with range dense in $\mathbf{D}_{\varepsilon}$ for all $\varepsilon \in (0, 1]$ then

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle_{B_{\varepsilon}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.36) $$

In particular for $\mathrm{cl}[(B_{\varepsilon})_{\varepsilon}] = 1$
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, \gamma \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.37) $$

Thus the normalized Colombeau integral with respect to a quadratic form $(\Delta_{\varepsilon})_{\varepsilon}$ reduces to the normalized integral when the quadratic form is the unit one on $\mathbf{H}$.

**▲ P.3.** If $B_{\varepsilon}^{-1}$ is bounded and everywhere defined for all $\varepsilon \in (0,1]$ then $\mathbf{D}_{\varepsilon} = \mathbf{H}$ with equivalent norms and $\Delta_{\varepsilon}(\alpha, \beta) = \langle \alpha, B_{\varepsilon}^{-1}\beta \rangle_{\mathbf{H}}$, $\forall \alpha, \beta \in \mathbf{H}$. Moreover $B_{\varepsilon}$ is self-adjoint, one has a direct sum decomposition $\mathbf{H} = \mathbf{H}_{+}^{\varepsilon} \oplus \mathbf{H}_{-}^{\varepsilon}$ of $\mathbf{H}$ and one has $B_{\varepsilon} = B_{+}^{\varepsilon} \oplus B_{-}^{\varepsilon}$ with $B_{+}^{\varepsilon}$ strictly positive, where $B_{\pm}^{\varepsilon}$ are the parts of $B_{\varepsilon}$ in $\mathbf{H}_{\pm}^{\varepsilon}$ resp. One has then the "Fubini theorem"

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} = \\ \left( \int_{\mathbf{H}_{+}^{\varepsilon}} \exp\left[\frac{i}{2} \|\gamma_1\|_{\mathbf{H}_{+}^{\varepsilon}}^2\right] \left\{ \int_{\mathbf{H}_{+}^{\varepsilon}} \exp\left[-\frac{i}{2} \|\gamma_2\|_{\mathbf{H}_{+}^{\varepsilon}}^2\right] \overline{\Phi_{\varepsilon}(\gamma_1 \oplus \gamma_2)} D[\gamma_2] \right\}^{-} D[\gamma_1] \right)_{\varepsilon} \quad (3.38) $$

where $\bar{\alpha}$ and $\{\cdot\}^{-}$ mean complex conjugates.

**▲ P.4.** If both $B_{\varepsilon}$ and $B_{\varepsilon}^{-1}$ are bounded everywhere on $\mathbf{H}$ for all $\varepsilon \in (0,1]$ then P.3 applies with $\mathbf{H} = \mathbf{D}_{\varepsilon}, \Delta_{\varepsilon} = B_{\varepsilon}$ and in this case

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, \gamma \rangle_{B_{\varepsilon}}\right] D[\gamma] \right)_{\varepsilon} = \\ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[-\frac{i}{2} \langle \alpha, B_{\varepsilon}\alpha \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon}. \quad (3.39) $$

**▲ P.5.** If $\dim \mathbf{H} < \infty$ i.e. $\mathbf{H} = \mathbb{R}^d$ then
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = (\mathbf{J}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, \gamma \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} \quad (3.37) $$

Thus the normalized Colombeau integral with respect to a quadratic form $(\Delta_{\varepsilon})_{\varepsilon}$ reduces to the normalized integral when the quadratic form is the unit one on $\mathbf{H}$.

**▲ P.3.** If $B_{\varepsilon}^{-1}$ is bounded and everywhere defined for all $\varepsilon \in (0,1]$ then $\mathbf{D}_{\varepsilon} = \mathbf{H}$ with equivalent norms and $\Delta_{\varepsilon}(\alpha, \beta) = \langle \alpha, B_{\varepsilon}^{-1}\beta \rangle_{\mathbf{H}}$, $\forall \alpha, \beta \in \mathbf{H}$. Moreover $B_{\varepsilon}$ is self-adjoint, one has a direct sum decomposition $\mathbf{H} = \mathbf{H}_{+}^{\varepsilon} \oplus \mathbf{H}_{-}^{\varepsilon}$ of $\mathbf{H}$ and one has $B_{\varepsilon} = B_{+}^{\varepsilon} \oplus B_{-}^{\varepsilon}$ with $B_{+}^{\varepsilon}$ strictly positive, where $B_{\pm}^{\varepsilon}$ are the parts of $B_{\varepsilon}$ in $\mathbf{H}_{\pm}^{\varepsilon}$ resp. One has then the "Fubini theorem"

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, B_{\varepsilon}\gamma \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\gamma) \right)_{\varepsilon} = \\ \left( \int_{\mathbf{H}_{+}^{\varepsilon}} \exp\left[\frac{i}{2} \|\gamma_1\|_{\mathbf{H}_{+}^{\varepsilon}}^2\right] \left\{ \int_{\mathbf{H}_{+}^{\varepsilon}} \exp\left[-\frac{i}{2} \|\gamma_2\|_{\mathbf{H}_{+}^{\varepsilon}}^2\right] \overline{\Phi_{\varepsilon}(\gamma_1 \oplus \gamma_2)} D[\gamma_2] \right\}^{-} D[\gamma_1] \right)_{\varepsilon} \quad (3.38) $$

where $\bar{\alpha}$ and $\{\cdot\}^{-}$ mean complex conjugates.

**▲ P.4.** If both $B_{\varepsilon}$ and $B_{\varepsilon}^{-1}$ are bounded everywhere on $\mathbf{H}$ for all $\varepsilon \in (0,1]$ then P.3 applies with $\mathbf{H} = \mathbf{D}_{\varepsilon}$, $\Delta_{\varepsilon} = B_{\varepsilon}$ and in this case

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2} \langle \gamma, \gamma \rangle_{B_{\varepsilon}}\right] D[\gamma] \right)_{\varepsilon} = \\ \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[-\frac{i}{2} \langle \alpha, B_{\varepsilon}\alpha \rangle_{\mathbf{H}}\right] d\mu_{\Phi_{\varepsilon}}(\alpha, B_{\varepsilon}) \right)_{\varepsilon} \quad (3.39) $$

**▲ P.5.** If $\dim \mathbf{H} < \infty$ i.e. $\mathbf{H} = \mathbb{R}^d$ then
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, \gamma\rangle_{B_{\varepsilon}}\right] D[\gamma] \right)_{\varepsilon} = (2\pi)^{-d/2} |\det(B_{\varepsilon})|_{\varepsilon} \left( \exp\left[-\frac{i\pi}{4} \mathbf{sign}(B_{\varepsilon})\right] \right)_{\varepsilon} \left( \int_{\mathbb{R}^d} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle\right] d\gamma \right)_{\varepsilon}, \quad (3.40) $$

where $\det B_{\varepsilon}$ and $\mathbf{sign}(B_{\varepsilon})$ are the determinant resp. the signature of $B_{\varepsilon}$.

▲ P.6. Invariance under translations by elements in $\mathbf{D}(B_{\varepsilon})$:

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle_{\mathbf{H}}\right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma + \alpha) \exp\left[\frac{i}{2}\langle\gamma + \alpha, B_{\varepsilon}(\gamma + \alpha)\rangle_{\mathbf{H}}\right] D[\gamma] \right)_{\varepsilon}. \quad (3.41) $$

## Appendix.IV. The Schrödinger equation with singular potential in $E_M[\mathcal{F}(\mathbb{R}^d)]$.

We consider the case of a Schrödinger operator of the form $-\Delta + (V_{\varepsilon}(x))_{\varepsilon}$, where $\Delta$ is the Laplacian on $\mathbb{R}^d$ and $(V_{\varepsilon}(x))_{\varepsilon}$ is a function in $E_M[\mathcal{F}(\mathbb{R}^d)]$. According to Feynman's heuristic considerations [13] we expect that the solution of the corresponding Colombeau-Schrödinger equation

$$ i \frac{\partial}{\partial t} (\Psi_{\varepsilon}(t, x))_{\varepsilon} = ([-\Delta + V_{\varepsilon}(x)] \Psi_{\varepsilon}(t, x))_{\varepsilon} \quad (4.1) $$

on the time interval $[0, T]$, with initial condition
$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}_{B_{\varepsilon}}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, \gamma\rangle_{B_{\varepsilon}}\right] D[\gamma] \right)_{\varepsilon} = (2\pi)^{-d/2} |\det(B_{\varepsilon})|_{\varepsilon} \left( \exp\left[-\frac{i\pi}{4} \mathbf{sign}(B_{\varepsilon})\right] \right)_{\varepsilon} \left( \int_{\mathbb{R}^d} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle\right] d\gamma \right)_{\varepsilon}, \quad (3.40) $$

where $\det B_{\varepsilon}$ and $\mathbf{sign}(B_{\varepsilon})$ are the determinant resp. the signature of $B_{\varepsilon}$.

**▲ P.6.** Invariance under translations by elements in $\mathbf{D}(B_{\varepsilon})$:

$$ (\mathbf{J}_{\Delta_{\varepsilon}}[\Phi_{\varepsilon}])_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma) \exp\left[\frac{i}{2}\langle\gamma, B_{\varepsilon}\gamma\rangle_{\mathbf{H}}\right] D[\gamma] \right)_{\varepsilon} = \left( \int_{\mathbf{H}} \Phi_{\varepsilon}(\gamma + \alpha) \exp\left[\frac{i}{2}\langle\gamma + \alpha, B_{\varepsilon}(\gamma + \alpha)\rangle_{\mathbf{H}}\right] D[\gamma] \right)_{\varepsilon}. \quad (3.41) $$

## Appendix.IV. The Schrödinger equation with singular potential in $E_M[\mathcal{F}(\mathbb{R}^d)]$.

We consider the case of a Schrödinger operator of the form $-\Delta + (V_{\varepsilon}(x))_{\varepsilon}$, where $\Delta$ is the Laplacian on $\mathbb{R}^d$ and $(V_{\varepsilon}(x))_{\varepsilon}$ is a function in $E_M[\mathcal{F}(\mathbb{R}^d)]$. According to Feynman's heuristic considerations [13] we expect that the solution of the corresponding Colombeau-Schrödinger equation

$$ i \frac{\partial}{\partial t} (\Psi_{\varepsilon}(t, x))_{\varepsilon} = \left( [-\Delta + V_{\varepsilon}(x)] \Psi_{\varepsilon}(t, x) \right)_{\varepsilon} \quad (4.1) $$

on the time interval $[0, T]$, with initial condition
$$ (\Psi_{\varepsilon}(0, x))_{\varepsilon} = (\varphi_{\varepsilon}(x))_{\varepsilon}. \qquad (4.2) $$

be given by a suitable normalized Colombeau path integral

$$ \int_{\gamma(T)=x} \varphi(\gamma(0)) \exp\left[\frac{i}{2}(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}\right] D[\gamma] \qquad (4.3) $$

where $(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}$ is the action correspondent to the given problem. In our problem the action is

$$ (S_{\varepsilon}^{T}(\gamma))_{\varepsilon} = \int_{0}^{T} \dot{\gamma}^{2}(t)dt - \int_{0}^{T} (V_{\varepsilon}(\gamma(t)))_{\varepsilon}dt \qquad (4.4) $$

and the integration should be in a suitable space of paths $\gamma$ ending at the instant $T$ in $x$. In order to be able to actually compute integrals of the form given via Eq.(4.3) It is natural to try to exploit the special feature of the phase function $(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}$, namely of consisting of a positive quadratic part plus the term involving $(V_{\varepsilon}(\gamma(t)))_{\varepsilon}$, considered as a perturbation. The most simple way then of taking advantage of the quadratic part as a "known" part is to introduce as an integration space a linear space of paths $\gamma$ for which $\int_{0}^{T} \dot{\gamma}^{2}(t)dt$ is finite i.e. a subset of all absolutely continuous paths. For simplicity it is natural to restrict the paths to those building a Hilbert space $\mathbf{H}$ in which $\|\gamma\| = (\int_{0}^{T} \dot{\gamma}^{2}(t)dt)^{1/2}$ is the norm and for which Fourier transforms are particularly easily computed, which is the case when the Fourier transform of $\exp[\frac{i}{2}\int_{0}^{T} \dot{\gamma}^{2}(t)dt]$ evaluated as $\Gamma(s, \cdot)$ is equal to $\exp[\frac{i}{2}\Gamma(s, s)]$, where $\Gamma(s, t) = T$-max($s, t$) is the kernel of $-d^2/dt^2$ with zero boundary condition e.g. at $t = T$ and zero derivative condition at $t = 0$. This then selects $\mathbf{H}$ to be the Hilbert space of absolutely continuous paths on $[0, T]$ of finite kinetic energy and being at the origin at time $t$ (we note that the space has to be at least as big as to contain $\Gamma(s, \cdot)$). Then we have for paths $\gamma \in \mathbf{H}$ that the expression (4.4) reads, for paths $\tilde{\gamma} := \gamma + x$ (with the correct property for (4.4) of ending at time $T$ in $x$)

$$ (S_{\varepsilon}^{T}(\tilde{\gamma}))_{\varepsilon} = (S_{\varepsilon}^{T}(\gamma + x))_{\varepsilon} = \|\gamma\|^2 - \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon}dt \qquad (4.4') $$
$$ (\Psi_{\varepsilon}(0, x))_{\varepsilon} = (\varphi_{\varepsilon}(x))_{\varepsilon}. \qquad (4.2) $$

be given by a suitable normalized Colombeau path integral

$$ \int_{\gamma(T)=x} \varphi(\gamma(0)) \exp\left[\frac{i}{2}(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}\right] D[\gamma] \qquad (4.3) $$

where $(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}$ is the action correspondent to the given problem. In our problem the action is

$$ (S_{\varepsilon}^{T}(\gamma))_{\varepsilon} = \int_{0}^{T} \dot{\gamma}^{2}(t)dt - \int_{0}^{T} (V_{\varepsilon}(\gamma(t)))_{\varepsilon}dt \qquad (4.4) $$

and the integration should be in a suitable space of paths $\gamma$ ending at the instant $T$ in $x$. In order to be able to actually compute integrals of the form given via Eq.(4.3) It is natural to try to exploit the special feature of the phase function $(S_{\varepsilon}^{T}(\gamma))_{\varepsilon}$, namely of consisting of a positive quadratic part plus the term involving $(V_{\varepsilon}(\gamma(t)))_{\varepsilon}$, considered as a perturbation. The most simple way then of taking advantage of the quadratic part as a "known" part is to introduce as an integration space a linear space of paths $\gamma$ for which $\int_{0}^{T} \dot{\gamma}^{2}(t)dt$ is finite i.e. a subset of all absolutely continuous paths. For simplicity it is natural to restrict the paths to those building a Hilbert space $\mathbf{H}$ in which $\|\gamma\| = (\int_{0}^{T} \dot{\gamma}^{2}(t)dt)^{1/2}$ is the norm and for which Fourier transforms are particularly easily computed, which is the case when the Fourier transform of $\exp[\frac{i}{2}\int_{0}^{T} \dot{\gamma}^{2}(t)dt]$ evaluated as $\Gamma(s, \cdot)$ is equal to $\exp[\frac{i}{2}\Gamma(s, s)]$, where $\Gamma(s, t) = T$-max($s, t$) is the kernel of $-d^{2}/dt^{2}$ with zero boundary condition e.g. at $t = T$ and zero derivative condition at $t = 0$. This then selects $\mathbf{H}$ to be the Hilbert space of absolutely continuous paths on $[0, T]$ of finite kinetic energy and being at the origin at time $t$ (we note that the space has to be at least as big as to contain $\Gamma(s, \cdot)$). Then we have for paths $\gamma \in \mathbf{H}$ that the expression (4.4) reads, for paths $\tilde{\gamma} := \gamma + x$ (with the correct property for (4.4) of ending at time $T$ in $x$)

$$ (S_{\varepsilon}^{T}(\tilde{\gamma}))_{\varepsilon} = (S_{\varepsilon}^{T}(\gamma + x))_{\varepsilon} = \|\gamma\|^2 - \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon}dt \qquad (4.4') $$
where $||\cdot||$ is the norm in $\mathbf{H}$. Thus the candidate for (4.3) becomes

$$ \int_{\gamma(T)=x} \varphi(\gamma(0)+x) \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp\left[-i \int_{0}^{T} (V_{\varepsilon}(\gamma(t)+x))_{\varepsilon} dt\right] D[\gamma]. \quad (4.5) $$

**Theorem 4.1** Assume $\varphi_{\varepsilon} \in E_M[\mathcal{F}(\mathbb{R}^d)]$, $V_{\varepsilon} \in E_M[\mathcal{F}(\mathbb{R}^d)]$, $\varepsilon \in (0,1]$ and define for $T \ge 0$

$$ ((\mathfrak{T}_T \varphi_{\varepsilon})(x))_{\varepsilon} = \int_{\mathbb{H}} \varphi(\gamma(0)+x) \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp\left[-i \int_{0}^{T} (V_{\varepsilon}(\gamma(t)+x))_{\varepsilon} dt\right] D[\gamma]. \quad (4.5') $$

Then for all $\varepsilon \in (0,1]$, $(\mathfrak{T}_T \varphi_{\varepsilon})(x)$ is a continuous bounded semigroup on $E_M[\mathcal{F}(\mathbb{R}^d)]$ with

$$ \|(\mathfrak{T}_T \varphi_{\varepsilon})(x)\|_{\mathcal{F}} \le \|\varphi_{\varepsilon}\|_{\mathcal{F}} \exp(t \|V_{\varepsilon}\|_{\mathcal{F}}), \quad (4.5'') $$

where $\|\cdot\|_{\mathcal{F}}$ is the norm on $E_M[\mathcal{F}(\mathbb{R}^d)]$.

**Proof.** Let $\gamma \in \mathbf{H}$ and write

$$ \gamma(t) = \gamma_1(t) + \gamma_2(t) \quad (4.6) $$

where $\gamma_2(t)$ is constant on the subinterval $[0, t_1]$ $0 < t_1 < t$ and $\gamma_2(t) = \gamma(t)$ on $[t_1, T]$, and where $\gamma_1(t) = 0$ on $[t_1, T]$, $\gamma_1(t) = \gamma_2(t) - \gamma_2(T)$ on $[0, t_1]$. Clearly (4.6) gives a splitting of $\mathbf{H}$ in the direct sum $\mathbf{H} = \mathbf{H}_1 \oplus \mathbf{H}_2$ where $\mathbf{H}_1$ consists of the paths $\gamma_1(t)$ and $\mathbf{H}_2$ consists of the paths $\gamma_2(t)$. By the property P5) of the normalized integral (Appendix III "Fubini Theorem") we have
where $||\cdot||$ is the norm in $\mathbf{H}$. Thus the candidate for (4.3) becomes

$$ \int_{\gamma(T)=x} \varphi(\gamma(0)+x) \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp\left[-i \int_{0}^{T} (V_{\varepsilon}(\gamma(t)+x))_{\varepsilon} dt\right] D[\gamma]. \quad (4.5) $$

**Theorem 4.1** Assume $\varphi_{\varepsilon} \in E_M[\mathcal{F}(\mathbb{R}^d)]$, $V_{\varepsilon} \in E_M[\mathcal{F}(\mathbb{R}^d)]$, $\varepsilon \in (0,1]$ and define for $T \ge 0$

$$ ((\mathfrak{T}_T \varphi_{\varepsilon})(x))_{\varepsilon} = \int_{\mathbb{H}} \varphi(\gamma(0)+x) \exp\left[\frac{i}{2} \|\gamma\|^2\right] \exp\left[-i \int_{0}^{T} (V_{\varepsilon}(\gamma(t)+x))_{\varepsilon} dt\right] D[\gamma]. \quad (4.5') $$

Then for all $\varepsilon \in (0,1]$, $(\mathfrak{T}_T \varphi_{\varepsilon})(x)$ is a continuous bounded semigroup on $E_M[\mathcal{F}(\mathbb{R}^d)]$ with

$$ \|(\mathfrak{T}_T \varphi_{\varepsilon})(x)\|_{\mathcal{F}} \le \|\varphi_{\varepsilon}\|_{\mathcal{F}} \exp(t \|V_{\varepsilon}\|_{\mathcal{F}}), \quad (4.5'') $$

where $\|\cdot\|_{\mathcal{F}}$ is the norm on $E_M[\mathcal{F}(\mathbb{R}^d)]$.

**Proof.** Let $\gamma \in \mathbf{H}$ and write

$$ \gamma(t) = \gamma_1(t) + \gamma_2(t) \quad (4.6) $$

where $\gamma_2(t)$ is constant on the subinterval $[0, t_1]$ $0 < t_1 < t$ and $\gamma_2(t) = \gamma(t)$ on $[t_1, T]$, and where $\gamma_1(t) = 0$ on $[t_1, T]$, $\gamma_1(t) = \gamma_2(t) - \gamma_2(T)$ on $[0, t_1]$. Clearly (4.6) gives a splitting of $\mathbf{H}$ in the direct sum $\mathbf{H} = \mathbf{H}_1 \oplus \mathbf{H}_2$ where $\mathbf{H}_1$ consists of the paths $\gamma_1(t)$ and $\mathbf{H}_2$ consists of the paths $\gamma_2(t)$. By the property P5) of the normalized integral (Appendix III "Fubini Theorem") we have
$$((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon =$$

$$\left( \tilde{\int}_{\mathbf{H}} \varphi(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon} =$$

$$\left( \tilde{\int}_{\mathbf{H}_1} \exp \left[ i \int_{t_1}^{T} \dot{\gamma}_2^2(t) dt \right] \exp \left[ -i \int_{t_1}^{T} (V_{\varepsilon}(\gamma_2(t) + x))_{\varepsilon} dt \right] \times \right. \quad (4.7)$$

$$\left\{ \tilde{\int}_{\mathbf{H}_1} \varphi(\gamma_1(0) + \gamma_2(t_1) + x) \exp \left[ i \int_{0}^{t_1} \dot{\gamma}_1^2(t) dt \right] \times \right.$$

$$\exp \left[ -i \int_{0}^{t_1} (V_{\varepsilon}(\gamma_1(t) + \gamma_2(t) + x))_{\varepsilon} dt \right] D[\gamma_1] \left\{ D[\gamma_2] \right\}_{\varepsilon} =$$

$$((\mathfrak{T}_{T-t_1}(\mathfrak{T}_{t_1} \varphi_{\varepsilon}))(x))_{\varepsilon},$$

which proves that $((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon$. is a semigroup. The continuity of $(\mathfrak{T}_T \varphi_\varepsilon)(x), \varepsilon \in (0,1]$ and the bound on its norm follow from the explicit computation of the normalized integral defining $(\mathfrak{T}_T \varphi_\varepsilon)(x)$ which is done by using the property PI (Appendix III) of the normalized integral i.e. that the Fresnel integral is a bounded linear functional on the Banach algebra $\mathcal{F}(\mathbf{H})$ so that
$$((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon =$$

$$\left( \tilde{\int}_{\mathbf{H}} \varphi(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon} =$$

$$\left( \tilde{\int}_{\mathbf{H}_1} \exp \left[ i \int_{t_1}^{T} \dot{\gamma}_2^2(t) dt \right] \exp \left[ -i \int_{t_1}^{T} (V_{\varepsilon}(\gamma_2(t) + x))_{\varepsilon} dt \right] \times \right. \quad (4.7)$$

$$\left\{ \tilde{\int}_{\mathbf{H}_1} \varphi(\gamma_1(0) + \gamma_2(t_1) + x) \exp \left[ i \int_{0}^{t_1} \dot{\gamma}_1^2(t) dt \right] \times \right.$$

$$\exp \left[ -i \int_{0}^{t_1} (V_{\varepsilon}(\gamma_1(t) + \gamma_2(t) + x))_{\varepsilon} dt \right] D[\gamma_1] \left\{ D[\gamma_2] \right\}_{\varepsilon} =$$

$$((\mathfrak{T}_{T-t_1}(\mathfrak{T}_{t_1} \varphi_\varepsilon))(x))_\varepsilon,$$

which proves that $((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon$. is a semigroup. The continuity of $(\mathfrak{T}_T \varphi_\varepsilon)(x), \varepsilon \in (0,1]$ and the bound on its norm follow from the explicit computation of the normalized integral defining $(\mathfrak{T}_T \varphi_\varepsilon)(x)$ which is done by using the property PI (Appendix III) of the normalized integral i.e. that the Fresnel integral is a bounded linear functional on the Banach algebra $\mathcal{F}(\mathbf{H})$ so that
$$ (\mathfrak{T}_T \varphi_\varepsilon)(x) = $$

$$ \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] = $$

$$ \sum_{n=0}^{\infty} \frac{(-i)^n}{n!} \underbrace{\int_0^T \dots \int_0^T}_{n} \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \left( \exp \left[ \frac{i}{2} \int_0^T \dot{\gamma}^2(t) dt \right] \prod_{j=1}^n V_{\varepsilon}(\gamma(t_j) + x) dt_j \right) D[\gamma] = \quad (4.8) $$

$$ \sum_{n=0}^{\infty} \frac{(-i)^n}{n!} \underbrace{\int_0^T \dots \int_0^T}_{n} \underbrace{\int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d}}_{n} \exp \left[ i \sum_{j=0, k=0} \alpha_{j, \varepsilon} \Gamma(t_j, t_k) \alpha_{k, \varepsilon} \right] \exp \left[ i \sum_{j=0} \alpha_{j, \varepsilon} \right] \times $$

$$ \prod_{j=1}^{n} (d\mu_{V}(\alpha_{j,\varepsilon}) dt_{j}) d\mu_{\phi_{\varepsilon}}(\alpha_{0,\varepsilon}), $$

where $\mu_{V_\varepsilon}$ and $\mu_{\phi_\varepsilon}$ are the measures on $\mathbb{R}^d$ of which $V_\varepsilon$ and $\phi_\varepsilon$ are Fourier transforms.

In fact Eq.(4.8) yields the bound

$$ \|(\mathfrak{T}_T \varphi_\varepsilon)(x)\|_{\mathcal{F}} \le \sum_{n=0}^{\infty} \frac{T^n}{n!} \|\mu_{V_\varepsilon}\| \|\mu_{\phi_\varepsilon}\| = \|\varphi_\varepsilon\|_{\mathcal{F}} \exp[T\|V_\varepsilon\|_{\mathcal{F}}]. $$

**Theorem 4.2.** The solution of the Schrödinger equation (4.1)-(4.2) with $V_\varepsilon$ and $\phi_\varepsilon$, $\varepsilon \in (0,1]$ in $\mathcal{F}(\mathbf{H})$ is given by $(\Psi_\varepsilon(T,x))_\varepsilon = ((\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon)(x))_\varepsilon$ where $((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon$ is given in Theorem 4.1 i.e. we have:

$$ ((\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon)(x))_\varepsilon = $$

$$ \left( \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^{2}(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon} \quad (4.9) $$

In the case where $(V_\varepsilon(\gamma))_\varepsilon$ is real the semigroup $\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon, t > 0, \varepsilon \in (0,1]$ is a semigroup mapping isometrically and bijectively $\mathcal{F}(\mathbb{R}^d) \cap L_2(\mathbb{R}^d)$ into itself, hence extends naturally to an unitary group $U_{T,\varepsilon}$ on $L_2(\mathbb{R}^d)$. In fact $U_{T,\varepsilon} = \exp[-iTH_\varepsilon]$ where $H_\varepsilon$ is the self-adjoint operator $-\Delta + V_\varepsilon, \varepsilon \in (0,1]$ in $L_2(\mathbb{R}^d)$.

**Proof:** It suffices to compare (4.8) with the computation of the convergent Dyson series for $U_{T,\varepsilon} = \exp[-iTH_\varepsilon]$.
$$ (\mathfrak{T}_T \varphi_\varepsilon)(x) = $$

$$ \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] = $$

$$ \sum_{n=0}^{\infty} \frac{(-i)^n}{n!} \underbrace{\int_0^T \dots \int_0^T}_{n} \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \left( \exp \left[ \frac{i}{2} \int_0^T \dot{\gamma}^2(t) dt \right] \prod_{j=1}^n V_{\varepsilon}(\gamma(t_j) + x) dt_j \right) D[\gamma] = \quad (4.8) $$

$$ \sum_{n=0}^{\infty} \frac{(-i)^n}{n!} \underbrace{\int_0^T \dots \int_0^T}_{n} \underbrace{\int_{\mathbb{R}^d} \dots \int_{\mathbb{R}^d}}_{n} \exp \left[ i \sum_{j=0, k=0} \alpha_{j, \varepsilon} \Gamma(t_j, t_k) \alpha_{k, \varepsilon} \right] \exp \left[ i \sum_{j=0} \alpha_{j, \varepsilon} \right] \times $$

$$ \prod_{j=1}^{n} (d\mu_{V}(\alpha_{j,\varepsilon}) dt_{j}) d\mu_{\phi_{\varepsilon}}(\alpha_{0,\varepsilon}), $$

where $\mu_{V_\varepsilon}$ and $\mu_{\phi_\varepsilon}$ are the measures on $\mathbb{R}^d$ of which $V_\varepsilon$ and $\phi_\varepsilon$ are Fourier transforms.

In fact Eq.(4.8) yields the bound

$$ \|(\mathfrak{T}_T \varphi_\varepsilon)(x)\|_{\mathcal{F}} \le \sum_{n=0}^{\infty} \frac{T^n}{n!} \|\mu_{V_\varepsilon}\| \|\mu_{\phi_\varepsilon}\| = \|\varphi_\varepsilon\|_{\mathcal{F}} \exp[T\|V_\varepsilon\|_{\mathcal{F}}]. $$

**Theorem 4.2.** The solution of the Schrödinger equation (4.1)-(4.2) with $V_\varepsilon$ and $\phi_\varepsilon$, $\varepsilon \in (0,1]$ in $\mathcal{F}(\mathbf{H})$ is given by $(\Psi_\varepsilon(T,x))_\varepsilon = ((\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon)(x))_\varepsilon$ where $((\mathfrak{T}_T \varphi_\varepsilon)(x))_\varepsilon$ is given in Theorem 4.1 i.e. we have:

$$ ((\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon)(x))_\varepsilon = $$

$$ \left( \int_{\mathbf{H}} \varphi_{\varepsilon}(\gamma(0) + x) \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon} \quad (4.9) $$

In the case where $(V_\varepsilon(\gamma))_\varepsilon$ is real the semigroup $\mathfrak{T}_{T,\varepsilon}\varphi_\varepsilon, t > 0, \varepsilon \in (0,1]$ is a semigroup mapping isometrically and bijectively $\mathcal{F}(\mathbb{R}^d) \cap L_2(\mathbb{R}^d)$ into itself, hence extends naturally to an unitary group $U_{T,\varepsilon}$ on $L_2(\mathbb{R}^d)$. In fact $U_{T,\varepsilon} = \exp[-iTH_\varepsilon]$ where $H_\varepsilon$ is the self-adjoint operator $-\Delta + V_\varepsilon, \varepsilon \in (0,1]$ in $L_2(\mathbb{R}^d)$.

**Proof:** It suffices to compare (4.8) with the computation of the convergent Dyson series for $U_{T,\varepsilon} = \exp[-iTH_\varepsilon]$.
**Remark 4.1.** The expression for the solution of Schrödinger's equation (4.1)-(4.2) is the realization of the formal Feymman expression (4.3) where the Feynman path $\gamma$ ending at time $t$ in $x$ is written as $\gamma := \gamma + x$ where $\gamma$ is a path in $\mathbf{H}$ i.e. ending at time $t$ in $0$. Actually the particular translation $\gamma \to \gamma + x$ could be replaced, by using the translation invariance of the normalized integral (property P4) of Appendix III), by any translation $\gamma \to \gamma + x + \gamma_0$ with $\gamma_0 \in \mathbf{H}$. The formulae in Theorems 4.1,4.2 would then take a different aspect (but would of course be equi-valent to the previous ones). In this sense the formula (4.3) can be understood as a short notation for a family of equivalent formulae, e.g, for the one given in Theorem 4.2.

**Remark 4.2.** The expression (4.8) for the solution of the Schrödinger equation (4.1), (4.2) can be put directly in relation with the one obtained from Varadhan and Maslov-Chebotarev's [27] way of expressing the Feynman-Colombeau integrals as integrals with respect to complex Colombeau-Poisson measures.
Let now $(G_{T,\varepsilon}(x,y))_{\varepsilon}$ be the Green's function for (4.1), (4.2) such that the solution of Colombeau-Schrödinger's equation (4.1),(4.2) is given by

$$ (\Psi_{\varepsilon}(T,x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \varphi_{\varepsilon}(y) G_{T,\varepsilon}(x,y) dy \right)_{\varepsilon} \qquad (4.9) $$

Using P.5 of Appendix III ("Fubini Theorem" for the normalized Colombeau integral) together with the splitting $\mathbf{H} = \mathbf{H}_0 \oplus \mathbf{H}_0^\perp$, with $\mathbf{H}_0 = \{y\eta(t)|y \in \mathbb{R}^d, \eta(t) = 1 - \frac{t}{T}\}$ we get the following:

**Theorem 4.3.** The Green's function for Colombeau-Schrödinger equation (4.1),(4.2) with $\varphi_{\varepsilon}, V_{\varepsilon} \in \mathcal{F}(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$ is given by

$$ (G_{T,\varepsilon}(x,y))_{\varepsilon} = G(T,x,y) \times \\ \left( \left( \int_{\mathbf{H}} \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \right) \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + (y-x)\eta(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon}. \qquad (4.10) $$

where

$$ G(T,x,y) = (2\pi i T)^{-d/2} \exp\left[i \frac{|x-y|^2}{2T}\right] \qquad (4.11) $$
**Remark 4.1.** The expression for the solution of Schrödinger's equation (4.1)-(4.2) is the realization of the formal Feymman expression (4.3) where the Feynman path $\gamma$ ending at time $t$ in $x$ is written as $\gamma := \gamma + x$ where $\gamma$ is a path in $\mathbf{H}$ i.e. ending at time $t$ in $0$. Actually the particular translation $\gamma \to \gamma + x$ could be replaced, by using the translation invariance of the normalized integral (property P4) of Appendix III), by any translation $\gamma \to \gamma + x + \gamma_0$ with $\gamma_0 \in \mathbf{H}$. The formulae in Theorems 4.1,4.2 would then take a different aspect (but would of course be equi-valent to the previous ones). In this sense the formula (4.3) can be understood as a short notation for a family of equivalent formulae, e.g, for the one given in Theorem 4.2.

**Remark 4.2.** The expression (4.8) for the solution of the Schrödinger equation (4.1), (4.2) can be put directly in relation with the one obtained from Varadhan and Maslov-Chebotarev's [27] way of expressing the Feynman-Colombeau integrals as integrals with respect to complex Colombeau-Poisson measures.
Let now $(G_{T,\varepsilon}(x,y))_{\varepsilon}$ be the Green's function for (4.1), (4.2) such that the solution of Colombeau-Schrödinger's equation (4.1),(4.2) is given by

$$ (\Psi_{\varepsilon}(T,x))_{\varepsilon} = \left( \int_{\mathbb{R}^d} \varphi_{\varepsilon}(y) G_{T,\varepsilon}(x,y) dy \right)_{\varepsilon} \qquad (4.9) $$

Using P.5 of Appendix III ("Fubini Theorem" for the normalized Colombeau integral) together with the splitting $\mathbf{H} = \mathbf{H}_0 \oplus \mathbf{H}_0^\perp$, with $\mathbf{H}_0 = \{y\eta(t)|y \in \mathbb{R}^d, \eta(t) = 1 - \frac{t}{T}\}$ we get the following:

**Theorem 4.3.** The Green's function for Colombeau-Schrödinger equation (4.1),(4.2) with $\varphi_{\varepsilon}, V_{\varepsilon} \in \mathcal{F}(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$ is given by

$$ (G_{T,\varepsilon}(x,y))_{\varepsilon} = G(T,x,y) \times \\ \left( \left( \int_{\mathbf{H}} \exp \left[ \frac{i}{2} \int_{0}^{T} \dot{\gamma}^2(t) dt \right] \right) \exp \left[ -i \int_{0}^{T} (V_{\varepsilon}(\gamma(t) + (y-x)\eta(t) + x))_{\varepsilon} dt \right] D[\gamma] \right)_{\varepsilon}. \qquad (4.10) $$

where

$$ G(T,x,y) = (2\pi i T)^{-d/2} \exp\left[i \frac{|x-y|^2}{2T}\right] \qquad (4.11) $$
is the Gireen's function in the case $V_\varepsilon = 0$.

Let us consider a Schrödinger-Colombeau operator of the form $-\Delta + (V_\varepsilon(x))_\varepsilon$, $x \in \mathbb{R}^d$ with $V_\varepsilon \in \mathcal{F}(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$ and real. Let $H_0 = \Delta$ and $H_\varepsilon = H_0 + V_\varepsilon(x)$, as self-adjoint operators on $L_2(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$. We consider now an operator $\exp(-itH_\varepsilon)\exp(itH_0)$ for $t \in \mathbb{R}$.

**Theorem 4.4.** Let $V_\varepsilon \in \mathcal{F}(\mathbb{R}^d)$ and $\varphi_\varepsilon \in \mathcal{F}(\mathbb{R}^d) \cap L_2(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$, then, for all $x \in \mathbb{R}^d$

$$ ([\exp(-itH_\varepsilon) \exp(itH_0) \varphi_\varepsilon](x))_\varepsilon = \left( \int_{\mathbb{R}^d} \exp(i\beta \cdot x) W_{t,\varepsilon}(x,\beta) d\mu_{\varphi_\varepsilon}(\beta) \right)_\varepsilon \quad (4.12) $$

where $\beta \cdot x$ is the scalar product in $\mathbb{R}^d$ and $\mu_{\varphi_\varepsilon}$ is the generalized Colombeau measure on $\mathbb{R}^d$ such that $(V_\varepsilon(x))_\varepsilon$ is the Colombeau-Fourier transform of $\mu_{\varphi_\varepsilon}$, and the quantity $(W_{t,\varepsilon}(x,\beta))_\varepsilon$ is given by the following normalized Colombeau paths integral:

$$ (W_{t,\varepsilon}(x,\beta))_\varepsilon = \left( \tilde{\int}_{H_-} \exp\left[ \frac{i}{2} \int_{-\infty}^0 \dot{\gamma}^2(\tau) d\tau \right] \exp\left[ -i \int_{-t}^0 V_\varepsilon(\gamma(\tau) + \beta\tau + x) d\tau \right] D[\gamma] \right)_\varepsilon \quad (4.13) $$

where $H_-$ is the Hllbert space of absolutely continuous paths $\gamma$ on the time interval $(-\infty,0]$, with values in $\mathbb{R}^d$, of finite kinetic energy $\int_{-\infty}^0 \dot{\gamma}^2(\tau) d\tau < \infty$

($\dot{\gamma}(\tau)$ is defined a.e., $\gamma$ being absolutely continuous) equal to the norm in $H_-$ and ending at time 0 at the origin.

For potentials such that the wave operators $W_{\pm,\varepsilon}$ exist for all $\varepsilon \in (0,1]$ as strong limits in $L_2(\mathbb{R}^d)$ of $\exp(-itH_\varepsilon)\exp(itH_0)$ as $t \to \pm\infty$ we have:
is the Gireen's function in the case $V_\varepsilon = 0$.

Let us consider a Schrödinger-Colombeau operator of the form $-\Delta + (V_\varepsilon(x))_\varepsilon$, $x \in \mathbb{R}^d$ with $V_\varepsilon \in \mathcal{F}(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$ and real. Let $H_0 = \Delta$ and $H_\varepsilon = H_0 + V_\varepsilon(x)$, as self-adjoint operators on $L_2(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$. We consider now an operator $\exp(-itH_\varepsilon)\exp(itH_0)$ for $t \in \mathbb{R}$.

**Theorem 4.4.** Let $V_\varepsilon \in \mathcal{F}(\mathbb{R}^d)$ and $\varphi_\varepsilon \in \mathcal{F}(\mathbb{R}^d) \cap L_2(\mathbb{R}^d)$ for all $\varepsilon \in (0,1]$, then, for all $x \in \mathbb{R}^d$

$$ ([\exp(-itH_\varepsilon) \exp(itH_0) \varphi_\varepsilon](x))_\varepsilon = \left( \int_{\mathbb{R}^d} \exp(i\beta \cdot x) W_{t,\varepsilon}(x,\beta) d\mu_{\varphi_\varepsilon}(\beta) \right)_\varepsilon \quad (4.12) $$

where $\beta \cdot x$ is the scalar product in $\mathbb{R}^d$ and $\mu_{\varphi_\varepsilon}$ is the generalized Colombeau measure on $\mathbb{R}^d$ such that $(V_\varepsilon(x))_\varepsilon$ is the Colombeau-Fourier transform of $\mu_{\varphi_\varepsilon}$, and the quantity $(W_{t,\varepsilon}(x,\beta))_\varepsilon$ is given by the following normalized Colombeau paths integral:

$$ (W_{t,\varepsilon}(x,\beta))_\varepsilon = $$

$$ \left( \tilde{\int}_{H_{-}} \exp \left[ \frac{i}{2} \int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau \right] \exp \left[ -i \int_{-t}^{0} V_{\varepsilon}(\gamma(\tau) + \beta\tau + x) d\tau \right] D[\gamma] \right)_{\varepsilon} \quad (4.13) $$

where $H_{-}$ is the Hllbert space of absolutely continuous paths $\gamma$ on the time interval $(-\infty, 0]$, with values in $\mathbb{R}^d$, of finite kinetic energy $\int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau < \infty$

($\dot{\gamma}(\tau)$ is defined a.e., $\gamma$ being absolutely continuous) equal to the norm in $H_{-}$ and ending at time 0 at the origin.

For potentials such that the wave operators $W_{\pm,\varepsilon}$ exist for all $\varepsilon \in (0,1]$ as strong limits in $L_2(\mathbb{R}^d)$ of $\exp(-itH_\varepsilon)\exp(itH_0)$ as $t \to \pm\infty$ we have:
$$ (W_{+,\epsilon}\varphi_{\epsilon})_{\epsilon} = \left( \int_{\mathbb{R}^d} \exp(iv_{-} \cdot x) W_{+,\epsilon}(x, v_{-}) d\mu_{\varphi_{\epsilon}}(v_{-}) \right)_{\epsilon} \quad (4.14) $$

with

$$ (W_{+,\epsilon}(x, v_{-}))_{\epsilon} = \left( \int_{H_{-}} \exp\left[ \frac{i}{2} \int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau \right] \exp\left[ -i \int_{-t}^{0} V_{\epsilon}(\gamma(\tau) + v_{-}\tau + x) d\tau \right] D[\gamma] \right)_{\epsilon} \quad (4.15) $$

where the integral is an "improper Colombeau-Fresnel integral" defined as the limit for

$t \to \infty$ of the Colombeau-Fresnel integral of the Colombeau-Fresnel integrable function:

$$ \exp\left[ -i \int_{-t}^{0} V_{\epsilon}(\gamma(\tau) + v_{-}\tau + x) d\tau \right]. $$

Similarly

$$ (W_{-,\epsilon}\varphi_{\epsilon})_{\epsilon} = \left( \int_{\mathbb{R}^d} \exp(-iv_{+} \cdot x) W_{-,\epsilon}(x, v_{+}) d\mu_{\varphi_{\epsilon}}(v_{+}) \right)_{\epsilon} \quad (4.16) $$

with

$$ (W_{-,\epsilon}(x, v_{-}))_{\epsilon} = \left( \int_{H_{+}} \exp\left[ \frac{i}{2} \int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau \right] \exp\left[ -i \int_{-t}^{0} V_{\epsilon}(\gamma(\tau) + v_{+}\tau + x) d\tau \right] D[\gamma] \right)_{\epsilon} \quad (4.17) $$

here $H_+$ is defined as $H_-$ with the time interval $(-\infty, 0]$ replaced by $[0, +\infty)$.
$$ (W_{+,\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(iv_{-} \cdot x) W_{+,\varepsilon}(x, v_{-}) d\mu_{\varphi_{\varepsilon}}(v_{-}) \right)_{\varepsilon} \quad (4.14) $$

with

$$ (W_{+,\varepsilon}(x, v_{-}))_{\varepsilon} = \left( \tilde{\int}_{H_{-}} \exp\left[ \frac{i}{2} \int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau \right] \exp\left[ -i \int_{-t}^{0} V_{\varepsilon}(\gamma(\tau) + v_{-}\tau + x) d\tau \right] D[\gamma] \right)_{\varepsilon} \quad (4.15) $$

where the integral is an "improper Colombeau-Fresnel integral" defined as the limit for

$t \to \infty$ of the Colombeau-Fresnel integral of the Colombeau-Fresnel integrable function:

$$ \exp\left[ -i \int_{-t}^{0} V_{\varepsilon}(\gamma(\tau) + v_{-}\tau + x) d\tau \right]. $$

Similarly

$$ (W_{-,\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = \left( \int_{\mathbb{R}^d} \exp(-iv_{+} \cdot x) W_{-,\varepsilon}(x, v_{+}) d\mu_{\varphi_{\varepsilon}}(v_{+}) \right)_{\varepsilon} \quad (4.16) $$

with

$$ (W_{-,\varepsilon}(x, v_{-}))_{\varepsilon} = \left( \tilde{\int}_{H_{+}} \exp\left[ \frac{i}{2} \int_{-\infty}^{0} \dot{\gamma}^2(\tau) d\tau \right] \exp\left[ -i \int_{-t}^{0} V_{\varepsilon}(\gamma(\tau) + v_{+}\tau + x) d\tau \right] D[\gamma] \right)_{\varepsilon} \quad (4.17) $$

here $H_+$ is defined as $H_-$ with the time interval $(-\infty, 0]$ replaced by $[0, +\infty)$.
**Remark 4.3.** In the case where $V_\epsilon$ is gentle for all $\epsilon \in (0, 1]$ such that the perturbation series in powers of $V_\epsilon$ for the wave operators converges for all $\epsilon \in (0, 1]$, the improper Colombeau-Fresnel integrals can also be defined by the termwise limit as $t \to \pm\infty$ of the convergent perturbation series obtained by expanding in powers of $V_\epsilon$ and interchanging the sum and the Colombeau-Fresnel Integral.

**Remark 4.4.** $(W_{+,\epsilon}(x, v_-))_\epsilon$ has e.g. the interpretation of the wave function at time 0 of a particle which had prescribed velocity $v_-$ at time $-\infty$.

# Appendix V. Generalized Perturbation of Dissipative Operators. Colombeau-Kato generalized Potentials.

**Definition 5.1.** By a contraction semigroup on a Banach space $\mathbb{R}$ we shall mean a family of bounded everywhere-defined linear operators $\mathbf{P}^t, 0 \le t < \infty$, mapping $\mathbb{R}$ into itself such that

$$ 
\begin{gathered}
\mathbf{P}^0 = 1, \mathbf{P}^t \mathbf{P}^s = \mathbf{P}^{t+s}, 0 \le t, s < \infty, \\
\|\mathbf{P}^t\| \le 1, 0 \le t < \infty, \qquad (5.1) \\
\lim_{t \to 0} \mathbf{P}^t \psi = \psi, \psi \in \mathbb{R}.
\end{gathered}
$$

**Definition 5.2.** The infinitesimal generator $\mathfrak{I}$ of a contraction semigroup $\mathbf{P}^t$ is defined by

$$ \mathfrak{I}\psi = \lim_{t \to 0} \left[ \frac{1}{t} (\mathbf{P}^t \psi - \psi) \right] \qquad (5.2) $$

on the domain $D(A)$ of all $\psi$ in $\mathbb{R}$ for which the limit exists.

**Definition 5.3.** Let $\mathbb{R}$ be a Banach space. By a contraction Colombeau semigroup on a Colombeau module $G_{\mathbb{R}}$ we shall mean a family of bounded everywhere-defined generalized linear operators $(\mathbf{P}_\epsilon^t)_{\epsilon \in (0,1]}$, $0 \le t < \infty$, mapping Colombeau module $G_{\mathbb{R}}$ into itself such that:
**Remark 4.3.** In the case where $V_\varepsilon$ is gentle for all $\varepsilon \in (0,1]$ such that the perturbation series in powers of $V_\varepsilon$ for the wave operators converges for all $\varepsilon \in (0,1]$, the improper Colombeau-Fresnel integrals can also be defined by the termwise limit as $t \to \pm\infty$ of the convergent perturbation series obtained by expanding in powers of $V_\varepsilon$ and interchanging the sum and the Colombeau-Fresnel Integral.

**Remark 4.4.** $(W_{+,\varepsilon}(x, v_-))_\varepsilon$ has e.g. the interpretation of the wave function at time 0 of a particle which had prescribed velocity $v_-$ at time $-\infty$.

# Appendix V. Generalized Perturbation of Dissipative Operators. Colombeau-Kato generalized Potentials.

**Definition 5.1.** By a contraction semigroup on a Banach space $\mathbb{R}$ we shall mean a family of bounded everywhere-defined linear operators $\mathbf{P}^t, 0 \le t < \infty$, mapping $\mathbb{R}$ into itself such that

$$ \mathbf{P}^0 = 1, \mathbf{P}^t \mathbf{P}^s = \mathbf{P}^{t+s}, 0 \le t, s < \infty, $$

$$ \|\mathbf{P}^t\| \le 1, 0 \le t < \infty, \qquad (5.1) $$

$$ \lim_{t \to 0} \mathbf{P}^t \psi = \psi, \psi \in \mathbb{R}. $$

**Definition 5.2.** The infinitesimal generator $\mathfrak{I}$ of a contraction semigroup $\mathbf{P}^t$ is defined by

$$ \mathfrak{I}\psi = \lim_{t \to 0} \left[ \frac{1}{t} (\mathbf{P}^t \psi - \psi) \right] \qquad (5.2) $$

on the domain $D(A)$ of all $\psi$ in $\mathbb{R}$ for which the limit exists.

**Definition 5.3.** Let $\mathbb{R}$ be a Banach space. By a contraction Colombeau semigroup on a Colombeau module $G_{\mathbb{R}}$ we shall mean a family of bounded everywhere-defined generalized linear operators $(\mathbf{P}_\varepsilon^t)_{\varepsilon \in (0,1]}$, $0 \le t < \infty$, mapping Colombeau module $G_{\mathbb{R}}$ into itself such that:
$$ (\mathbf{P}_{\varepsilon}^{0})_{\varepsilon} = 1, $$

$$ (\mathbf{P}_{\varepsilon}^{t})\mathbf{P}_{\varepsilon}^{s} = (\mathbf{P}_{\varepsilon}^{t}\mathbf{P}_{\varepsilon}^{s})_{\varepsilon} = (\mathbf{P}^{t+s})_{\varepsilon}, \quad 0 \le t, s < \infty, $$

$$ (\|\mathbf{P}_{\varepsilon}^{t}\|)_{\varepsilon} \le 1, 0 \le t < \infty, \qquad (5.3) $$

$$ \left( \lim_{t \to 0} \mathbf{P}_{\varepsilon}^{t} \psi \right)_{\varepsilon} = \psi, \psi \in \mathbb{R}, $$

$$ \varepsilon \in (0, 1]. $$

**Definition 5.4.** The infinitesimal generator $(\mathfrak{T}_{\varepsilon})_{\varepsilon}$ of a contraction semigroup $(\mathbf{P}_{\varepsilon}^{t})_{\varepsilon}$ is
defined by

$$ ((\mathfrak{T}_{\varepsilon})_{\varepsilon})\psi = \left( \lim_{t \to 0} \left[ \frac{1}{t} (\mathbf{P}_{\varepsilon}^{t} \psi - \psi) \right] \right)_{\varepsilon} \qquad (5.4) $$

on the domain $\mathbf{D}(\mathfrak{T})$ of all $\psi$ in $\mathbb{R}$ for which the limit exists.

**Definition 5.5.** We shall call an operator $A$ on a Banach space $\mathbb{R}$ dissipative in case for all $\psi$ in $\mathbf{D}(A)$, if $\varphi$ is in the space $\mathbb{R}^*$ of continuous linear functionals on $\mathbb{R}, \|\varphi\| = 1$ and $(\psi, \varphi) = \|\psi\|$ then $\mathrm{Re}(A\psi, \varphi) \le 0$.

**Definition 5.6.**

**Theorem 5.1.** [41]. Let $\mathfrak{T}$ be the infinitesimal generator of a contraction semigroup on the Banach space $\mathbb{R}$ and let $B$ be a dissipative operator with $\mathbf{D}(B) \supset \mathbf{D}(\mathfrak{T})$. If there exist constants $a$ and $b$ with $a < 1/2$ such that for all $\psi$ in $\mathbf{D}(\mathfrak{T})$

$$ \|B\psi\| \le a\|\mathfrak{T}\| + b\|\psi\|, \qquad (5.5) $$

then $\mathfrak{T} + B$ is the infinitesimal generator of a contraction semigroup.

**Theorem 5.2.** [41].
$$ (\mathbf{P}_{\varepsilon}^{0})_{\varepsilon} = 1, $$

$$ (\mathbf{P}_{\varepsilon}^{t})\mathbf{P}_{\varepsilon}^{s} = (\mathbf{P}_{\varepsilon}^{t}\mathbf{P}_{\varepsilon}^{s})_{\varepsilon} = (\mathbf{P}^{t+s})_{\varepsilon}, \quad 0 \le t, s < \infty, $$

$$ (\|\mathbf{P}_{\varepsilon}^{t}\|)_{\varepsilon} \le 1, 0 \le t < \infty, \qquad (5.3) $$

$$ \left( \lim_{t \to 0} \mathbf{P}_{\varepsilon}^{t} \psi \right)_{\varepsilon} = \psi, \psi \in \mathbb{R}, $$

$$ \varepsilon \in (0, 1]. $$

**Definition 5.4.** The infinitesimal generator $(\mathfrak{T}_{\varepsilon})_{\varepsilon}$ of a contraction semigroup $(\mathbf{P}_{\varepsilon}^{t})_{\varepsilon}$ is
defined by

$$ ((\mathfrak{T}_{\varepsilon})_{\varepsilon})\psi = \left( \lim_{t \to 0} \left[ \frac{1}{t} (\mathbf{P}_{\varepsilon}^{t} \psi - \psi) \right] \right)_{\varepsilon} \qquad (5.4) $$

on the domain $\mathbf{D}(\mathfrak{T})$ of all $\psi$ in $\mathbb{R}$ for which the limit exists.

**Definition 5.5.** We shall call an operator $A$ on a Banach space $\mathbb{R}$ dissipative in case for all $\psi$ in $\mathbf{D}(A)$, if $\varphi$ is in the space $\mathbb{R}^*$ of continuous linear functionals on $\mathbb{R}, \|\varphi\| = 1$ and $(\psi, \varphi) = \|\psi\|$ then $\mathrm{Re}(A\psi, \varphi) \le 0$.

**Definition 5.6.**

**Theorem 5.1.** [41]. Let $\mathfrak{T}$ be the infinitesimal generator of a contraction semigroup on the Banach space $\mathbb{R}$ and let $B$ be a dissipative operator with $\mathbf{D}(B) \supset \mathbf{D}(\mathfrak{T})$. If there exist constants $a$ and $b$ with $a < 1/2$ such that for all $\psi$ in $\mathbf{D}(\mathfrak{T})$

$$ \|B\psi\| \le a\|\mathfrak{T}\| + b\|\psi\|, \qquad (5.5) $$

then $\mathfrak{T} + B$ is the infinitesimal generator of a contraction semigroup.

**Theorem 5.2.** [41].
$$ \pi^{-1} I_v(z) = \int_0^\pi e^{z\cos\theta} \cos(v\theta) d\theta - \sin(v\pi) \int_0^\infty e^{-z(cht)-vt} dt $$

$$ I_{\sqrt{2ME}}\left(\frac{M\omega u_a u_b}{i \sin(\omega S)}\right), z = \frac{M\omega u_a u_b}{i \sin(\omega S)}, v = \sqrt{2ME} $$

## References

[1] J.F. Colombeau, “New Generalized Functions and Multiplication of Distributions,” (North Holland, Amsterdam, 1984).

[2] J.F. Colombeau, “Elementary Introduction to New Generalized Functions,” (North Holland, Amsterdam 1985).

[3] M.Grosser, E.Farkas, M.Kunzinger, R.Steinbauer, On the foundations of nonlinear generalized functions I,II,Mem.Am.Math.Soc.,153, 729, (2001).

[4] M.Oberguggenberger, M. Grosser, M.Kunzinger, Gunther Hormann, Nonlinear Theory of Generalized Functions (Research Notes in Mathematics Series) 400 pages. Publisher: Chapman and Hall/CRC; 1 edition, March 16, 1999.

[5] H.Biagioni, Nonlinear theory of generalized functions. Springer, 1990. ISBN: 3540524088, Pages: 226.

[6] H. Vernaevel, “Ideals in the Ring of Colombeau Generalized Numbers,” Communications in Algebra, Volume 38, Issue 6, 2010.

[7] J.Aragona, A.Ronaldo, G.Garcia, S.O.Juriaans, Colombeau's Algebra of full Generalized Numbers. http://arxiv.org/abs/0909.0348

[8] J.Aragoaa, S.O.Juriaansa, O.R.B.Oliveira and D.Scarpalezosa, Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions. Proceedings of the Edinburgh Mathematical Society (Series 2) (2008), 51: 545-564.

[9] C.Garetto, “Pseudodifferential operators with generalized symbols and regularity theory.” Master thesis. http://www2.imperial.ac.uk/~cgaretto/thesis.pdf

[10] M.Kunzinger, M.Oberguggenberger, R.Steinbauer, J.A.Vickers, Generalized Flows and Singular ODE's on Differentiable Manifolds. Acta Applicandae Mathematicae Volume 80, Number 2, 221-241.

[11] M.Nedeljkov and S.Pilipović. “Convolution in Colombeau's spaces of generalized functions,” parts I and II. Publ. Inst. math. (Beograd), 52(66):
$$ \pi^{-1} I_v(z) = \int_0^\pi e^{z\cos\theta} \cos(v\theta) d\theta - \sin(v\pi) \int_0^\infty e^{-z(cht)-vt} dt $$

$$ I_{\sqrt{2ME}}\left(\frac{M\omega u_a u_b}{i \sin(\omega S)}\right), z = \frac{M\omega u_a u_b}{i \sin(\omega S)}, v = \sqrt{2ME} $$

## References

[1] J.F. Colombeau, "New Generalized Functions and Multiplication of Distributions," (North Holland, Amsterdam, 1984).

[2] J.F. Colombeau, "Elementary Introduction to New Generalized Functions," (North Holland, Amsterdam 1985).

[3] M.Grosser, E.Farkas, M.Kunzinger, R.Steinbauer, On the foundations of nonlinear generalized functions I,II,Mem.Am.Math.Soc.,153, 729, (2001).

[4] M.Oberguggenberger, M. Grosser, M.Kunzinger, Gunther Hormann, Nonlinear Theory of Generalized Functions (Research Notes in Mathematics Series) 400 pages. Publisher: Chapman and Hall/CRC; 1 edition, March 16, 1999.

[5] H.Biagioni, Nonlinear theory of generalized functions. Springer, 1990. ISBN: 3540524088, Pages: 226.

[6] H. Vernaevel, Deals in the Ring of Colombeau Generalized Numbers. Communications in Algebra, Volume 38, Issue 6, 2010.

[7] J.Aragona, A.Ronaldo, G.Garcia, S.O.Juriaans, Colombeau's Algebra of full Generalized Numbers. http://arxiv.org/abs/0909.0348

[8] J.Aragoaa, S.O.Juriaansa, O.R.B.Oliveira and D.Scarpalezosa, Algebraic and Geometric Theory of the Topological Ring of Colombeau Generalized Functions. Proceedings of the Edinburgh Mathematical Society (Series 2) (2008), 51: 545-564.

[9] C.Garetto, Pseudodifferential operators with generalized symbols and regularity theory. Master thesis. http://www2.imperial.ac.uk/~cgaretto/thesis.pdf

[10] M.Kunzinger, M.Oberguggenberger, R.Steinbauer, J.A.Vickers, Generalized Flows and Singular ODE's on Differentiable Manifolds. Acta Applicandae Mathematicae Volume 80, Number 2, 221-241.

[11] M.Nedeljkov and S.Pilipović. Convolution in Colombeau's spaces of generalized functions, parts I and II. Publ. Inst. math. (Beograd), 52(66):
[12] M.Oberguggenberger.Pseudodifferential operators on algebras of generalized functions: elliptic regularity. Unpublished manuscript, lectures at University of Pretoria and École Normale Supérieure de Lyon, 1990.

[13] R.P.Feynman, A.R.Hibbs, "Quantum Mechanics and Path Integrals". R.P. Feynman, Space time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367-387.

[14] H.Kleinert and A.Chervyakov, Simple Explicit Formulas for Gaussian Path Integrals with Time-Dependent Frequencies. Physics Letters A, Volume 245, Issue 5, 24 August 1998, Pages 345-357.

[15] H.Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific Publishing Co., Singapore 1995, Second extended edition, pp.1-850.

[16] I.M.Gelfand and A.M.Yaglom, J.Math.Phys.1,48 (1960).

[17] H.Ahmedov, I.H.Duru, A.E.Gumrukcuoglu, A Simple Path Integration For the Time Dependent Oscillator. http://arxiv.org/abs/quant-ph/0511095

[18] Dae-Yup Song, Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential. Physical Review A 62 (2000) 014103. http://arxiv.org/abs/quant-ph/0002065

[19] M.Blau, Notes on (Semi-)Advanced Quantum Mechanics: The Path Integral Approach to Quantum Mechanics. http://www.blau.itp.unibe.ch/lecturesPI.pdf

[20] V.P.Maslov, Complex Markov Cheins and Continual Feynman Integral. Nauka Moskov 1976.

[21] V.P.Maslov, Theory of Perturbations and Asimptotic Methods. Izd.MGU, Moscow, (in Russian) 1965.

[22] V.P. Maslov, M.V.Fedoriuk, Semiclassical Approximations in Quantum Mechanics. D. Reidel, Dordrecht, Holland, 1981. ix+301pp.

[23] V.P.Maslov, M.V.Fedorjuk, Logarithmic asymptotics of fast decaying solutions to Petrovskii type hyperbolic systems, Math.Notes,v.45,1989, N 5, pp.50-62

[24] V.P.Maslov, Operational Methods, "Mir", 1976.

[25] V.P.Maslov, The quasi-classical asymptotic solutions of some problems in mathematical physics. I. Computational Mathematics and Mathematical Physics, Volume 1, Issue 1, 1961, Pages 113–128.

[26] V.P. Maslov, The quasi-classical asymptote to the solutions of some problems in mathematical physics. II. Computational Mathematics and Mathematical Physics, Volume 1, Issue 3, 1962, Pages 744-778.

[27] V.P.Maslov, "Stationary-phase method for Feynman's continual integral", Theoret. and Math. Phys., 2:1 (1970), pp.21–25.
[12] M.Oberguggenberger.Pseudodifferential operators on algebras of generalized functions: elliptic regularity. Unpublished manuscript, lectures at University of Pretoria and École Normale Supérieure de Lyon, 1990.

[13] R.P.Feynman, A.R.Hibbs, "Quantum Mechanics and Path Integrals". R.P. Feynman, Space time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367-387.

[14] H.Kleinert and A.Chervyakov, Simple Explicit Formulas for Gaussian Path Integrals with Time-Dependent Frequencies. Physics Letters A, Volume 245, Issue 5, 24 August 1998, Pages 345-357.

[15] H.Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific Publishing Co., Singapore 1995, Second extended edition, pp.1-850.

[16] I.M.Gelfand and A.M.Yaglom, J.Math.Phys.1,48 (1960).

[17] H.Ahmedov, I.H.Duru, A.E.Gumrukcuoglu, A Simple Path Integration For the Time Dependent Oscillator. http://arxiv.org/abs/quant-ph/0511095

[18] Dae-Yup Song, Unitary relation between a harmonic oscillator of time-dependent frequency and a simple harmonic oscillator with and without an inverse-square potential. Physical Review A 62 (2000) 014103. http://arxiv.org/abs/quant-ph/0002065

[19] M.Blau, Notes on (Semi-)Advanced Quantum Mechanics: The Path Integral Approach to Quantum Mechanics. http://www.blau.itp.unibe.ch/lecturesPI.pdf

[20] V.P.Maslov, Complex Markov Cheins and Continual Feynman Integral. Nauka Moskov 1976.

[21] V.P.Maslov, Theory of Perturbations and Asimptotic Methods. Izd.MGU, Moscow, (in Russian) 1965.

[22] V.P. Maslov, M.V.Fedoriuk, Semiclassical Approximations in Quantum Mechanics. D. Reidel, Dordrecht, Holland, 1981. ix+301pp.

[23] V.P.Maslov, M.V.Fedorjuk, Logarithmic asymptotics of fast decaying solutions to Petrovskii type hyperbolic systems, Math.Notes,v.45,1989, N 5, pp.50-62

[24] V.P.Maslov, Operational Methods, "Mir", 1976.

[25] V.P.Maslov, The quasi-classical asymptotic solutions of some problems in mathematical physics. I. Computational Mathematics and Mathematical Physics, Volume 1, Issue 1, 1961, Pages 113–128.

[26] V.P. Maslov, The quasi-classical asymptote to the solutions of some problems in mathematical physics. II. Computational Mathematics and Mathematical Physics, Volume 1, Issue 3, 1962, Pages 744-778.

[27] V.P.Maslov, "Stationary-phase method for Feynman's continual integral", Theoret. and Math. Phys., 2:1 (1970), pp.21–25.
V.P.Maslov,A.M.Chebotarev,Theoret. and Math.Phys.,28:3 (1976) pp.291-306.

[28] E.B.Ipatov,D.S.Lukin,E.A.Palkin,The numerical realization of Maslov's canonical operator method in problems of HF-wave propagation in the earth's ionosphere.Radiofizika (ISSN 0021-3462),Vol.33,No.5, May 1990, pp.562-572.

[29] E.B.Ipatov,D.S.Lukin,E.A.Palkin,Maslov canonical operator in problems of numerical simulation of diffraction and propagation of waves in inhomogeneous media.Russian Journal of Numerical Analysis and Mathematical Modelling.Vol.5,No.6,pp.465-488.

[30] S.Dobrokhotov,S.Sekerzh-Zenkovich,B.Tirozzi, T.Tudorovskiy, Generalized Maslov canonical operator and tsunami asymptotics over nonuniform bottom.I.arXiv:math-ph/0601001v1

[31] A.Bermüdez, D. Gomez, J.E.Roberts. Fifth International Conference on Mathematical and Numerical Aspects of Wave 2000 - Science -1038 p.

[32] M.B.Mensky, Quantum Measurements and Decoherence. Models and Phenomenology. Kluwer Academic Publishers, Dordrecht / Boston / London, 2000. vol.110 in the series Fundamental Theories of Physics. ISBN 0-7923-6227-6, Russian translation: Fizmatlit, Moscow, 2001.

[33] M.B.Mensky,Continuous Quantum Measurements and Path Integrals. IOP Publishing, Bristol and Philadelphia, 1993 [in English; Japanese expanded translation: Yoshioka, Kyoto, 1995]

[34] M.B.Mensky, Decoherence and the theory of continuous quantum measurements,Uspekhi Fiz.Nauk 168,1017-1035 (1998) in Russian, English translation in Physics-Uspekhi 41, 923-940 (1998). PDF-file: Mensky in Physics-Uspekhi 1998, v.41, no.9 quant-ph/9812017

[35] M.B.Mensky, Continuously Measured Systems, Path Integrals and Information, Intern. J. Theor. Phys. 37, 273-280 (1998) quant-ph/0007089

[36] S.M.Buckley, Sobolev Spaces. http://eprints.nuim.ie/1604/1/BuckleySobolev27.pdf

[37] V.G.Bagrov,V.V.Belov and I.M.Ternov,Quasiclassical trajectory-coherent states of a nonrelativistic particle in an arbitrary electromagnetic field.Theoretical and Mathematical Physics, 1982, 50:3, 256-261.

[38] S.Albeverio,Ph.Combe,R.Høegh-Krohn, G.Rideau,M.Sirugue-Collin, M.Sirugue,R.Stora,Feynman path integrals. (Proc. Marseille, 1978), Lecture Notes in Phys.,vol.106, Springer, Berlin,1979, pp.3-57.

[39] S.A.Albeverio and R.J.Høegh-Krohn,Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics, vol. 523, Springer-Verlag, Berlin, 1976.

[40] D.Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math.J.Volume 47, Number 3 (1980), 559-600.

[41] E.Nelson,Feynman integrals and the Schrödinger equation,J.Math.Phys.5
V.P.Maslov,A.M.Chebotarev,Theoret. and Math.Phys.,28:3 (1976) pp.291-306.

[28] E.B.Ipatov,D.S.Lukin,E.A.Palkin,The numerical realization of Maslov's canonical operator method in problems of HF-wave propagation in the earth's ionosphere.Radiofizika (ISSN 0021-3462),Vol.33,No.5, May 1990, pp.562-572.

[29] E.B.Ipatov,D.S.Lukin,E.A.Palkin,Maslov canonical operator in problems of numerical simulation of diffraction and propagation of waves in inhomogeneous media.Russian Journal of Numerical Analysis and Mathematical Modelling.Vol.5,No.6,pp.465-488.

[30] S.Dobrokhotov,S.Sekerzh-Zenkovich,B.Tirozzi, T.Tudorovskiy, Generalized Maslov canonical operator and tsunami asymptotics over nonuniform bottom.I.arXiv:math-ph/0601001v1

[31] A.Bermüdez, D. Gomez, J.E.Roberts. Fifth International Conference on Mathematical and Numerical Aspects of Wave 2000 - Science -1038 p.

[32] M.B.Mensky, Quantum Measurements and Decoherence. Models and Phenomenology. Kluwer Academic Publishers, Dordrecht / Boston / London, 2000. vol.110 in the series Fundamental Theories of Physics. ISBN 0-7923-6227-6, Russian translation: Fizmatlit, Moscow, 2001.

[33] M.B.Mensky,Continuous Quantum Measurements and Path Integrals. IOP Publishing, Bristol and Philadelphia, 1993 [in English; Japanese expanded translation: Yoshioka, Kyoto, 1995]

[34] M.B.Mensky, Decoherence and the theory of continuous quantum measurements,Uspekhi Fiz.Nauk 168,1017-1035 (1998) in Russian, English translation in Physics-Uspekhi 41, 923-940 (1998). PDF-file: Mensky in Physics-Uspekhi 1998, v.41, no.9 quant-ph/9812017

[35] M.B.Mensky, Continuously Measured Systems, Path Integrals and Information, Intern. J. Theor. Phys. 37, 273-280 (1998) quant-ph/0007089

[36] S.M.Buckley, Sobolev Spaces. http://eprints.nuim.ie/1604/1/BuckleySobolev27.pdf

[37] V.G.Bagrov,V.V.Belov and I.M.Ternov,Quasiclassical trajectory-coherent states of a nonrelativistic particle in an arbitrary electromagnetic field.Theoretical and Mathematical Physics, 1982, 50:3, 256-261.

[38] S.Albeverio,Ph.Combe,R.Høegh-Krohn, G.Rideau,M.Sirugue-Collin, M.Sirugue,R.Stora,Feynman path integrals. (Proc. Marseille, 1978), Lecture Notes in Phys.,vol.106, Springer, Berlin,1979, pp.3-57.

[39] S.A.Albeverio and R.J.Høegh-Krohn,Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics, vol. 523, Springer-Verlag, Berlin, 1976.

[40] D.Fujiwara, Remarks on convergence of the Feynman path integrals, Duke Math.J.Volume 47, Number 3 (1980), 559-600.

[41] E.Nelson,Feynman integrals and the Schrödinger equation,J.Math.Phys.5
(1964), pp.332–343.

[42] M.Stojanovir'c, Regularization for heat kernel in nonlinear paraboloc equations, Taiwanese Journal of Mathematics, Vol.12, No.1, pp.63-87, February 2008.

[43] M.Nedeljkov and D.Rajter-Ćirić, Generalized uniformly continuous semigroups and semilinear hyperbolic systems with regularized derivatives, Monatshefte Für Mathematik, Volume 160, Number 1, 81-93.

[44] Z.Haba, Semiclassical stochastic representation of the Feynman integral, J.Phys. A. Math. Gen. 27 (1994) 6457-6477.

[45] Z.Haba, Probabilistic representation of quantum dynamics, Physics Letters A, 175 (1993) 371-376.

[46] L.C.Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998.

[47] L.Schwartz, Theorie des distributions, 2 ed. Hermann, Paris, 1966.

[48] V.N.Kolokoltsov, Complex Measures on Path Space: An Introduction to the Feynman Integral Applied to the Schrödinger Equation, Methodology And Computing In Applied Probability, October 1999, Volume 1, Issue 3, pp. 349-365.

[49] V.N.Kolokoltsov, Mathematics of the Feynman path integral, Talk presented at the IMC "FILOMAT 2001", Nis, August 26-30, 2001, FILOMAT 2001. http://silver.ima.umn.edu/prob-pde/reprints-preprints/kolokoltsov/serbia.pdf

[50] V.N.Kolokoltsov, Markov Processes, Semigroups and Generators, Studies in Mathematics 38. March 17, 2011, ISBN-10: 3110250101, ISBN-13: 978-3110250107, Edition 1.

[51] E.J.McShane, Integrals devised for special purposes, Bull. Amer. Math. Soc. 69 (1963),597-627.

[52] R.H.Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math, and Phys. 39 (1960), 126-140.

[53] Yu.Yu.Lobanov and V.D.Rushai, Development of the Green function method on a basis of deterministic approach to approximate functional integration. http://www1.jinr.ru/Proceedings/Burdik-2005/pdf/lobanov.pdf http://www1.jinr.ru/Preprints/2003/113(P11-2003-113).pdf

[54] E.M.Stein and R.Shakarchi, Real analysis: measure theory, integration, and Hilbert spaces. 2005.

[55] C.Garetto, M.Oberguggenberger, Generalised Fourier integral operator methods for hyperbolic equations with singularities, http://arxiv.org/abs/1104.3035v1

[56] Y.G. Wang, Semilinear geometric optics for generalized solutions. Vienna, Preprint ESI 829 (2000) http://www.mat.univie.ac.at/~esiprpr/esi829.pdf

[57] D. J. Griffiths, Introduction to Quantum Mechanics, 2e. Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 106–109. (2005). ISBN 0131118927.

[58] J. von Neumann (1932). Mathematische Grundlagen der Quantenmechanik.
(1964), pp.332–343.

[42] M.Stojanovir'c, Regularization for heat kernel in nonlinear paraboloc
equations, Taiwanese Journal of Mathematics, Vol.12, No.1, pp.63-87,
February 2008.

[43] M.Nedeljkov and D.Rajter-Ćirić, Generalized uniformly continuous semigroups and semilinear hyperbolic systems with regularized derivatives, Monatshefte Für Mathematik, Volume 160, Number 1, 81-93.

[44] Z.Haba, Semiclassical stochastic representation of the Feynman integral, J.Phys. A. Math. Gen.27 (1994) 6457-6477.

[45] Z.Haba, Probabilistic representation of quantum dynamics, Physics Letters A, 175 (1993) 371-376.

[46] L.C.Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998.

[47] L.Schwartz, Theorie des distributions, 2 ed. Hermann, Paris, 1966.

[48] V.N.Kolokoltsov, Complex Measures on Path Space: An Introduction to the Feynman Integral Applied to the Schrödinger Equation, Methodology And Computing In Applied Probability, October 1999, Volume 1, Issue 3, pp. 349-365.

[49] V.N.Kolokoltsov, Mathematics of the Feynman path integral, Talk presented at the IMC "FILOMAT 2001", Nis, August 26-30, 2001, FILOMAT 2001.
http://silver.ima.umn.edu/prob-pde/reprints-preprints/kolokoltsov/serbia.pdf

[50] V.N.Kolokoltsov, Markov Processes, Semigroups and Generators, Studies in Mathematics 38. March 17, 2011, ISBN-10: 3110250101, ISBN-13: 978-3110250107, Edition 1.

[51] E.J.McShane, Integrals devised for special purposes, Bull. Amer. Math. Soc. 69 (1963),597-627.

[52] R.H.Cameron, A family of integrals serving to connect the Wiener and Feynman integrals, J. Math, and Phys. 39 (1960),126-140.

[53] Yu.Yu.Lobanov and V.D.Rushai, Development of the Green function method on a basis of deterministic approach to approximate functional integration.
http://www1.jinr.ru/Proceedings/Burdik-2005/pdf/lobanov.pdf
http://www1.jinr.ru/Preprints/2003/113(P11-2003-113).pdf

[54] E.M.Stein and R.Shakarchi, Real analysis:measure theory,integration,and Hilbert spaces. 2005.

[55] C.Garetto, M.Oberguggenberger, Generalised Fourier integral operator methods for hyperbolic equations with singularities, http://arxiv.org/abs/1104.3035v1

[56] Y.G. Wang, Semilinear geometric optics for generalized solutions. Vienna, Preprint ESI 829 (2000)
http://www.mat.univie.ac.at/~esiprpr/esi829.pdf

[57] D. J. Griffiths, Introduction to Quantum Mechanics, 2e. Upper Saddle River, New Jersey: Pearson Prentice Hall. pp. 106–109. (2005).
ISBN 0131118927.

[58] J. von Neumann (1932). Mathematische Grundlagen der Quantenmechanik.
Berlin: Springer. (German)

J. von Neumann (1955). Mathematical Foundations of Quantum Mechanics.

Princeton University Press. (English)

[59] M. Schlosshauer, (2005). "Decoherence, the measurement problem, and interpretations of quantum mechanics". Rev. Mod. Phys. 76 (4):1267-1305.
arXiv:quant-ph/0312059.

[60] W. H. Zurek, (2009). "Quantum Darwinism". Nature Physics 5: 181-188.
arXiv:0903.5082

[61] N.G. van Kampen, Physica A 153, 97 (1988).

[62] A. Hobson, Two-photon interferometry and quantum state collapse,
Phys. Rev. A 88, 022105 – Published 8 August 2013

[63] S. Weinberg, Phys. Rev. A 85, 062116 (2012).

[64] A. Bassi, K. Lochan, S. Satin, T. P. Singh, H. Ulbricht, Models of
wave function Collapse, Underlying Theories, and Experimental Tests,
Rev. Mod. Phys. 85, 471-527 (2013) http://arxiv.org/abs/1204.4325

[65] K.Jacobs, D. A. Steck, A Straightforward Introduction to Continuous Quantum Measurement, Contemporary Physics 47, 279 (2006)
DOI:10.1080/00107510601101934

[66] M.B.Mensky, Continuous Quantum Measurements and Path Integrals,
Hardcover: 188 pp. Publisher: CRC Press (January 1, 1993)
ISBN-10: 0750302283 ISBN-13: 978-0750302289

[67] M.B.Mensky, Quantum Measurements and Decoherence: Models and Phenomenology, Series: Fundamental Theories of Physics, Vol. 110
2000, XVI, 231 pp. ISBN 978-94-015-9566-7

[68] A.J. Leggett, Schrödinger's Cat and Her Laboratory Cousins, Contemp.
Phys., 1984, v.25, No.6, pp. 583-598.

[69] G. Gamow, "Zur Quantentheorie des Atomkernes", Z. Physik 51, 204 (1928)

[70] R. Feinman, A. Hibbs, (2005), Quantum Mechanics and Path Integrals, Dover (Emended Edition).

[71] A. Bassi, K. Lochan, S. Satin, T. P. Singh, H. Ulbricht, Models of
wave function Collapse, Underlying Theories, and Experimental Tests,
Rev. Mod. Phys. 85, 471-527 (2013) http://arxiv.org/abs/1204.4325

[72] K.Jacobs, D. A. Steck, A Straightforward Introduction to Continuous Quantum Measurement, Contemporary Physics 47, 279 (2006)
DOI:10.1080/00107510601101934

[73] E. Schrödinger (1935) "Die gegenwärtige Situation in der Quantenmechanik (The present situation in quantum mechanics)".
Naturwissenschaften 23 (49): 807-812.doi:10.1007/BF0491891.

[74] H.Wimmel (1992). Quantum physics & observed reality: a critical interpretation of quantum mechanics. World Scientific. p. 2. ISBN 978-981-02-1010-6.

[75] Faye, J (2008-01-24). "Copenhagen Interpretation of Quantum Mechanics".
Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab Center for