Basic non-Archimedean functional analysis over
non-Archimedean field

★ \_c\#\_ . *Applicationtoconstructivequantumfieldtheory*

jaykov foukzon

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jaykov foukzon. Basic non-Archimedean functional analysis over non-Archimedean field
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# Non-Archimedean functional analysis over non-Archimedean field $\mathbb{R}_c^#$. Applications to constructive quantum field theory.

Jaykov Foukzon

Center for Mathematical Sciences, Israel Institution of Technology, Haifa, Israel.

E-mail: jaykovfoukzon@list.ru

E-mail: jaikovfoukzon@gmail.com

https://orcid.org/0000-0002-6837-4004

**Abstract.** Functional analysis works with TVS (Topological Vector Spaces), classically over archimedean fields like $\mathbb{R}$ and $\mathbb{C}$. Canonical non-Archimedean functional analysis, where alternative but equally valid number systems such as p-adic numbers $\mathbb{Q}_p$ etc. are fundamental, is a fast-growing discipline.

This paper deals with TVS over non-classical non-Archimedean fields $\mathbb{R}_c^#$ and $\mathbb{C}_c^#$. Definitions and theorems related to non-Archimedean functional analysis on non-Archemedean field $\mathbb{R}_c^#$ and on complex field $\mathbb{C}_c^# = \mathbb{R}_c^# + i\mathbb{R}_c^#$ are considered. Applications to constructive quantum field theory also are considered [6]. Definitions and theorems appropriate to analysis on non-Archemedean field $\mathbb{R}_c^#$ and on complex field $\mathbb{C}_c^# = \mathbb{R}_c^# + i\mathbb{R}_c^#$ are given in [1]-[2].

Content.

## Chapter 0. Introduction

§1. Bivalent hyper Infinitary first-order logic $^2L_{\infty}^{#}$ with restricted rules of conclusion. Generalized Deduction Theorem.

§2. Set theory $NC_{\infty}^{#}$.

§3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^2L_{\infty}^{#}$ with restricted canonical rules of conclusion

§4. Internal Set Theory IST.

§5. Internal Set Theory IST#.

§6. Hypernaturals $N^#$.

§7. Axioms of the nonstandard arithmetic $A^#$.

§8. The Generalized Recursion Theorem.

§9. General associatíve and commutative laws.

§10. Hyperrationals $\mathbb{Q}^#$.

§11. External Cauchy hyperreals $\mathbb{R}_c^#$ via Cauchy completion of $\mathbb{Q}^#$.

## Chapter I. $\mathbb{R}_c^$-Valued abstract measures and integration.

§1. $\sigma$#-Algebras

§2. $\mathbb{R}_c^$-Valued $\#$-measures

§2.1. $\#$-Convergence of functions and the generalized Egoroff theorem.

§2.2. Vector-valued $\#$-measures

§3. The Lebesgue $\#$-Integral
§ 4. #-Convergence in #-measure.

§ 5. The Extension of #-Measure

§ 5.1. Outer #-measures.

§ 5.2. The Lebesgue and Lebesgue-Stieltjes #-measure on *$\mathbb{R}_c^#$.

§ 5.3. Product #-measures.

§ 5.4. Lebesgue #-measure and Lebesgue #-integral in *$\mathbb{R}_c^{#n}$.

§ 5.5. Lebesgue #-integrable functions on *$\mathbb{R}_c^{#n}$.

**Chapter II.** *$\mathbb{R}_c^#$-valued distribution.

§1. *$\mathbb{R}_c^#$-valued test functions and distributions

§ 2. The non-Archimedean external $*C_c^#$-Valued Schwartz distributions.

§ 2.1. Schwartz space $S^#(*\mathbb{R}_c^{#n})$.

§ 2.2. Schwartz space $S_{fin}^#(*\mathbb{R}_{c,fin}^{#n})$

§ 2.3. *$C_c^#$-Valued tempered distributions.

§ 3. The Fourier transform on $S^#(*\mathbb{R}_c^{#n})$, $S_{fin}^#(*\mathbb{R}_c^{#n})$

**Chapter III. Non-Archimedean Hilbert Spaces over field *$C_c^#$**

§ 1. Hilbert Spaces over field *$C_c^#$. Basics.

§ 2. Non-Archimedean Hilbert Space Basis.

§ 3. #-Analytic vectors. Generalized Nelson's #-analytic vector theorem.

§ 4. The generalized Spectral Theorem.

§ 4.1. The #-continuous functional calculus related to bounded in *$\mathbb{R}_c^#$ self-#-adjoint operators.

§ 4.2. The spectral #-measures.

§ 4.2.1. Multiplicity free operators.

§ 4.2.2. Operators of uniform multiplicity.

§ 4.2.3. Disjoint #-measure classes.

§ 4.3. Spectral projections.

§ 4.4. The #-continuous functional calculus related to unbounded in *$\mathbb{R}_c^#$ self-#-adjoint operators.

§ 4.5. Nearstandard $C_c^*$ algebras generated by spectral projections related to unbounded in *$\mathbb{R}_c^#$ self-#-adjoint operators.

§ 4.6. *$C_c^#$-valued quadratic forms.

§ 4.7. #-Convergence of unbounded in *$\mathbb{R}_c^#$ operators.

§ 4.8. Graph #-limits.

§ 4.9. The polar decomposition.

§ 4.10. Generalized Trotter product formula.

§ 5. Tensor products and second quantization.

§ 5.1. Tensor products.

§ 5.2. Non-Archimedean Fock spaces.

§ 5.3. Second quantization of the free Hamiltonian.

**Chapter IV.** Non-Archimedean Banach spaces endowed with *$\mathbb{R}_c^#$-valued norm.

§1. Definitions and examples.

§2. Linear operators, isomorphisms.

**Chapter V.** Semigroups of operators on a non-Archimedean Banach spaces.

§1. Semigroups on non-Archimedean Banach spaces and their generators.

§2. Hypercontractive semigroups.

**Chapter VI.** Singular Perturbations of Selfadjoint Operators on a non-Archimedean
Hilbert space.

§1.Introduction.

§2. Strong #-Convergence of Operators.

§3. Estimates on a G #-Convergent hyper infinite Sequence.

§ 4.Estimates for singular perturbations.

**Chapter VII.**

§1.Free scalar field.

§2. Q#-space representation of the non-Archimedean Fock space structures.

**Chapter VIII. A non-Archemedean Banach algebras and C#-Algebras**

§1.A non-Archemedean Banach algebra B(H#)

§1.1.Basic Properties.

§1.2 Types of Operators.

§1.3.Basic Spectral Theory.

§1.4. $l_2^#(G)$ and $B(l_2^#(G))$

§1.5.Topologies on B(H#).

§2.Non-Archemedean Banach algebras and C#-Algebras.

§2.1. Initial Definitions and #-Continious Functional Calculus.

§2.2. *C#-valued States.

§2.3.Representations and the generalized Gelfand-Naimark-Segal Construction.

Introduction

The incompleteness of set theory ZFC leads one to look for natural nonconservative
extensions of ZFC in which one can prove statements independent of ZFC which
appear to be “true”. One approach has been to add large cardinal axioms.

Or, one can investigate second-order expansions like Kelley-Morse class theory,
KM or

Tarski-Grothendieck set theory TG or It is a nonconservative extension of ZFC and is
obtained from other axiomatic set theories by the inclusion of Tarski's axiom which
implies the existence of inaccessible cardinals. See also related set theory with a filter
quantifier ZF(aa). In this paper we look at a set theory $NC_{\infty}^{#}$ [18], based on bivalent
gyper infinitary logic with restricted Modus Ponens Rule [18]. Nonconservative
extension namely IST# of the canonical internal set theory IST was presented in [18].

§1.Bivalent hyper Infinitary first-order logic $^{2}L_{\infty}^{#}$ with
restricted rules of conclusion.Generalized Deduction
Theorem.

Hyper infinitary language $L_{\infty}^{#}$ are defined according to the length of hyper infinitary
conjunctions/disjunctions as well as quantification it allows. In that way, assuming a
supply of $\kappa < \aleph_0^{#} = \text{card}(\aleph^{#})$ variables to be interpreted as ranging over a nonempty
domain, one includes in the inductive definition of formulas an infinitary clause for
conjunctions and disjunctions, namely, whenever the hypernaturals indexed hyper
infinite sequence $\{A_{\delta}\}_{\delta \in \aleph^{#}}$ of formulas has length less than $\kappa$, one can form the
hyperfinite conjunction/disjunction of them to produce a formula. Analogously, whenever
an hypernaturals indexed sequence of variables has length less than $\lambda$, one can
introduce one of the quantifiers $\forall$ or $\exists$ together with the sequence of variables in front of
a formula to produce a new formula. One also stipulates that the length of any
well-formed formula is less than $\aleph_0^#$ itself.

The syntax of bivalent hyper infinitary first-order logics $^{2}L_{\infty}^{\#}$ consists of a (ordered) set of sorts and a set of function and relation symbols, these latter together with the

corresponding type, which is a subset with less than $\aleph_0^# = \text{card}(\mathbb{N}^#)$ many sorts.
Therefore, we assume that our signature may contain relation and function symbols on
$\gamma < \aleph_0^#$ many variables, and we suppose there is a supply of $\kappa < \aleph_0^#$ many fresh
variables of each sort. Terms and atomic formulas are defined as usual, and general
formulas are defined inductively according to the following rules.

If $\phi, \psi, \{\phi_\alpha : \alpha < \gamma\}$ (for each $\gamma < \kappa$) are formulas of $L_{\infty,\#}^+$, the following are also formulas:

(i) $\bigwedge_{\alpha<\gamma} \phi_{\alpha}, \bigwedge_{\alpha\leq\gamma} \phi_{\alpha},$

(ii) $\bigvee_{\alpha<\gamma} \phi_{\alpha}, \bigvee_{\alpha\leq\gamma} \phi_{\alpha},$

$$
(iii) \phi \rightarrow \psi, \phi \land \psi, \phi \lor \psi, \neg \phi
$$

$$
(iv) \forall_{\alpha<\gamma} x_{\alpha} \phi \text{ (also written } \forall \mathbf{x}_{\gamma} \phi \text{ if } \mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}),
$$

$$
(v) \exists_{\alpha<\gamma} x_{\alpha} \phi \text{ (also written } \exists \mathbf{x}_{\gamma} \phi \text{ if } \mathbf{x}_{\gamma} = \{x_{\alpha} : \alpha < \gamma\}),
$$

$$
\begin{align*}
& (\text{vi}) \text{ the statement } \bigwedge_{\alpha<\gamma} \phi_{\alpha} \text{ holds if and only if for any } \alpha \text{ such that } \alpha < \gamma \\
& \text{the statement holds } \phi_{\alpha},
\end{align*}
$$

$$
\begin{align*}
& \text{(vii) the statement } \bigvee_{\alpha<\gamma} \phi_{\alpha} \text{ holds if and only if there exist } \alpha \text{ such that } \alpha < \gamma \\
& \text{the statement holds } \phi_{\alpha}.
\end{align*}
$$

**Definition 1.1.** A valuation of a syntactic system is a function that as signs $\vdash$ (true)
to some of its sentences, and/or $\perp$ (false) to some of its sentences. Precisely, a
valuation maps a nonempty subset of the set of sentences into the set $\{\vdash, \perp\}$.
We call a valuation bivalent iff it maps all the sentences into $\{\vdash, \perp\}$.

**Definition 1.2.** Let $L$ be a propositional language. $L$ is a classical bivalent propositional
language iff its admissible valuations are the functions $v$ such that for all sentences
$A, B$

of $L$ the following properties hold

(a) $v(A) \in \{\top, \bot\}$

$$
(b) v(\neg A) = \top \text{ iff } v(A) = \bot
$$

$$
(c) v(A \land B) = \top \text{ iff } v(A) = v(B) = \top.
$$

(d) by definition of the classical implication $A \Rightarrow B$ the following truth table holds

$v(A) \quad v(B) \quad v(A \Rightarrow B)$

(1) $\top \quad \top \quad \top$

(2) $\top \quad \bot \quad \bot$

(3) $\bot \quad \top \quad \top$

(4) $\bot \quad \bot \quad \top$

Truth table 1.

$$
(e) v^*(A) \in \{\top, \bot\}
$$

$$
(f) v^*(\neg A) = \top \text{ iff } v^*(A) = \bot
$$

$$
(g) v^*(A \wedge B) = \top \text{ iff } v^*(A) = v^*(B) = \top.
$$
(h) by definition of the nonclassical implication $A \Rightarrow B$ the following truth table holds

$$v^*(A) \quad v^*(B) \quad v^*(A \Rightarrow B)$$

(1) $\top$ $\top$ $\top$

(2) $\top$ $\bot$ $\top$

(3) $\bot$ $\top$ $\top$

(4) $\bot$ $\bot$ $\top$

Truth table 2.

**Remark 1.1.** Note that in the case (2) of the truth table 2

$$\top = v^*(A \Rightarrow B) \neq v(A \Rightarrow B) = \bot.$$

In this case we call implication $A \Rightarrow B$ a weak implication and abbreviate

$$A \Rightarrow_B B \qquad (1.1)$$

We call a statement (1.1) as a weak statement and often abbreviate $v(A \Rightarrow B) = \top_w$ instead (1).

**Definition 1.3.** [7-8]. A is a valid (logically valid) sentence (in symbols, $\models A$) in $L$ iff every admissible valuation of $L$ satisfies A.

The axioms of hyper infinitary first-order logic $^{2}L_{\infty}^{\#}$ consist of the following schemata:

**I. Logical axiom**

A 1. $A \rightarrow [B \rightarrow A]$

A 2. $[A \rightarrow [B \rightarrow C] \rightarrow [[A \rightarrow B] \rightarrow [A \rightarrow C]]]$

A 3. $[\neg B \rightarrow \neg A] \rightarrow [A \rightarrow B]$

A 4. $[\bigwedge_{i<\alpha} [A \rightarrow A_i]] \rightarrow [A \rightarrow \bigwedge_{i<\alpha} A_i], \alpha \in \mathbb{N}^{\#}$

A 5. $[\bigwedge_{i<\alpha} A_i] \rightarrow A_j, \alpha \in \mathbb{N}^{\#}$

A 6. $[\forall \mathbf{x}[A \rightarrow B] \rightarrow [A \rightarrow \forall \mathbf{x}B]]$

provided no variable in $\mathbf{x}$ occurs free in $A$;

A 7. $\forall \mathbf{x}A(\mathbf{x}) \rightarrow S_f(A)$,

where $S_f(A)$ is a substitution based on a function $f$ from $\mathbf{x}$ to the terms of the language; in particular:

A 7'. $\forall x_i[A(x_i)] \Rightarrow A(t)$ is a wff of $^{2}L_{\infty}^{\#}$ and $t$ is a term of $^{2}L_{\infty}^{\#}$ that is free for $x_i$ in $A(x_i)$. Note here that $t$ may be identical with $x_i$; so that all wffs $\forall x_i A \Rightarrow A$ are axioms by virtue of axiom (7), see [8].

A 8. Gen (Generalization).

$\forall x_i B$ follows from $B$.

**II. Restricted rules of conclusion.**

Let $\mathcal{F}_{wfr}$ be a set of the all closed wffs of $L_{\infty}^{\#}$.

**R1.** RMP (Restricted Modus Ponens).

There exist subsets $\Delta_1, \Delta_2 \subset \mathcal{F}_{wfr}$ such that the following rules are satisfied.

From $A$ and $A \Rightarrow B$, conclude $B$ iff $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$, where $\Delta_1, \Delta_2 \subset \mathcal{F}_{wfr}$.

In particular for any $A, B \in \mathcal{F}_{wfr}: A \Rightarrow_B B \in \Delta_2$.

If $A \notin \Delta_1$ and $(A \Rightarrow B) \notin \Delta_2$ we also abbreviate by $A, A \Rightarrow_B B \vdash_{RMP} B$.

**R2.** RMT (Restricted Modus Tollens)

There exist subsets $\Delta'_1, \Delta'_2 \subset \mathcal{F}_{wfr}$ such that the following rules are satisfied.

$$P \Rightarrow Q, \neg Q \vdash_{RMT} \neg P \text{ iff } P \notin \Delta'_1 \text{ and } (P \Rightarrow Q) \notin \Delta'_2, \text{ where } \Delta'_1, \Delta'_2 \subset \mathcal{F}_{wfr}.$$

**Remark 1.2.** Note that RMP and RMT easily prevent any paradoxes of naive Cantor set theory (NC), see [1],[9].
**III. Additional derived rule of conclusion.**

**Particularization rule (RPR)**

Remind that canonical unrestricted particularization rule (UPR) reads

**UPR:** If **t** is free for *x* in *B*(*x*), then ∀*x*[*B*(*x*)] ⊢ *B*(**t**), see [8].

**Proof.** From ∀x[B(x)] and the instance ∀x[B(x)] ⇒ B(t) of axiom (A7), we obtain B(t)
by unrestricted modus ponens rule. Since *x* is free for *x* in *B*(*x*), a special case of
unrestricted particularization rule is: ∀xB ⊢ B.

**Definition 1.4.** Any formal theory *L* with a hyper infinitary lenguage *L*<sub>∞</sub><sup>#</sup> is defined when the following conditions are satisfied:

1. A hyper infinite set of symbols is given as the symbols of L. A finite or hyperfinite sequence of symbols of L is called an expression of L.

2. There is a subset of the set of expressions of *L* called the set of *well formed formulas (wffs) of* *L*. There is usually an effective procedure to determine whether a given expression is a wff.

3. There is a set of wfs called the set of axioms of L. Most often, one can effectively decide whether a given wff is an axiom; in such a case, L is called an axiomatic theory.

4. There is a *finite* set $R_1, \dots, R_n$, of relations among wffs, called rules of conclusion. For each $R_i$, there is a unique positive integer $j$ such that, for every set of $j$ wfs and each wff $B$, one can effectively decide whether the given $j$ wffs are in the relation $R_i$ to $B$, and, if so, $B$ is said to follow from or to be a direct consequence of the given wffs by virtue of $R_j$.

**Definition 1.5.** A proof in *L* is a finite or *hyperfinite* sequence *B*₁, ..., *B*ₖ, *k* ∈ ℵ<sup>#</sup> of wffs such that for each *i*, either *B*ᵢ is an axiom of *L* or *B*ᵢ is a direct consequence of some of the preceding wffs in the sequence by virtue of one of the rules of inference of *L*.

**Definition 1.6.** A theorem of *L* is a wff *B* of *Y* such that *B* is the last wff of some proof in *L*. Such a proof is called a proof of *B* in *L*.

**Definition 1.7.** A wff *E* is said to be a consequence in *L* of a set of Γ of wffs if and only if there is a finite or *hyperfinite* sequence *B*₁, ..., *B*ₖ, *k* ∈ ℵ<sup>#</sup> of wffs such that *E* is *B*ₖ and, for each *i*, either *B*ᵢ is an axiom or *B*ᵢ is in Γ, or *B*ᵢ is a direct consequence by some rule of inference of some of the preceding wffs in the sequence. Such a sequence is colled a proof (or deduction) *E* from Γ. The members of Γ are called the hypotheses or premisses of the proof.
We use Γ ⊢ *E* as an abbreviation for *E* as a consequence of Γ.
In order to avoid confusion when dealing with more than one theory, we write Γ ⊢<sub>*L*</sub> *E*, adding the subscript *L* to indicate the theory in question.
If Γ is a finite or *hyperfinite* set {*H*ᵢ}<sub>1≤i≤m</sub>, *m* ∈ ℵ<sup>#</sup> we write *H*₁, ..., *H*ₚ ⊢ *E* instead of {*H*ᵢ}<sub>1≤i≤m</sub> ⊢ *E*.

**Lemma 1.1.** [18]. ⊢ B ⇒ B for all wffs B.

**Theorem 1.1.** (Generalized Deduction Theorem 1). If Γ is a set of wffs and B and E are wffs, and Γ, B ⊢ E, then Γ ⊢ B ⇒<sub>s</sub> E. In particular, if B ⊢ E then ⊢ B ⇒ E.

**Proof.** Let $E_1, \dots, E_n, n \in \mathbb{N}^#$ be a proof of *E* form $\Gamma \cup \{B\}$, where $E_n$ is *E*.
Let us prove, by *hyperfinite* induction on *j*, that $\Gamma \vdash B \Rightarrow_s E_j$ for $1 \le j \le n$.
First of all, $E_1$ must be either in $\Gamma$ or an axiom of *L* or *B* itself.
By axiom schema A1, $E_1 \Rightarrow_s (B \Rightarrow_s E_1)$ is an axiom. Hence, in the first two cases, by MP, $\Gamma \vdash B \Rightarrow_s E_1$. For the third case, when $E_1$ is $B$, we have $\vdash B \Rightarrow_s E_1$ by Lemma 1, and, therefore, $\Gamma \vdash B \Rightarrow_s E_1$. This takes care of the case $j = 1$.

Assume now that: $\vdash B \Rightarrow_s E_k$ for all $k < j, j \in \mathbb{N}^#$. Either $E_j$ is an axiom, or $E_j$ is in $\Gamma$, or $E_j$ is $B$, or $E_j$ follows by modus ponens from some $E_l$ and $E_m$ where $l < j, m < j$, and $E_m$ has the form $E_l \Rightarrow_s E_j$. In the first three cases, $\Gamma \vdash B \Rightarrow_s E_j$ as in the case $j = 1$ above. In the last case, we have, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s E_l$ and $\Gamma \vdash B \Rightarrow_s (E_l \Rightarrow_s E_j)$. But, by axiom schema (A2), $\vdash B \Rightarrow_s (E_l \Rightarrow_s E_j) \Rightarrow_s ((B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j))$

Hence, by MP, $\Gamma \vdash (B \Rightarrow_s E_l) \Rightarrow_s (B \Rightarrow_s E_j)$ and, again by MP, $\Gamma \vdash B \Rightarrow_s E_j$. Thus, the proof by hyperfinite induction is complete.

The case $j = n \in \mathbb{N}^#$ is the desired result. Notice that, given a deduction of $E$ from $\Gamma$ and $B$, the proof just given enables us to construct a deduction of $B \Rightarrow_s E$ from $\Gamma$. Also note that axiom schema A3 was not used in proving the generalized deduction theorem.

**Remark 1.3.** For the remainder of the chapter, unless something is said to the contrary,

we shall omit the subscript $L$ in $\vdash_L$. In addition, we shall use $\Gamma, B \vdash E$ to stand for $\Gamma \cup \{B\} \vdash E$. In general, we let $\Gamma, B_1, \dots, B_n \vdash E$ stand for $\Gamma \cup \{B_i\}_{1 \le i \le n} \vdash E$.

**Remark 1.4.** We shall use the terminology proof, theorem, consequence, axiomatic, etc. and notation $\Gamma \vdash E$ introduced above.

**Proposition 1.1.** Every wff $B$ of $K$ that is an instance of a tautology is a theorem of $K$, and it may be proved using only axioms A1-A3 and MP.

**Proposition 1.2.** If $E$ does not depend upon $B$ in a deduction showing that $\Gamma, B \vdash E$, then $\Gamma \vdash E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$, in which $E$ does not depend upon $B$. In this deduction, $D_n$ is $E$. As an inductive hypothesis, let us assume that the proposition is true for all deductions of length less than $n \in \mathbb{N}^#$. If $E$ belongs to $\Gamma$ or is an axiom, then $\Gamma \vdash E$. If $E$ is a direct consequence of one or two preceding wffs by Gen or MP, then, since $E$ does not depend upon $B$, neither do these preceding wfs. By the inductive hypothesis, these preceding wfs are deducible from $\Gamma$ alone. Consequently, so is $E$.

**Theorem 1.2.** (Generalized Deduction Theorem 2). Assume that, in some deduction showing that $\Gamma, B \vdash E$, no application of Gen to a wff that depends upon $B$ has as its quantified variable a free variable of $B$. Then $\Gamma \vdash B \Rightarrow_s E$.

**Proof.** Let $D_1, \dots, D_n$ be a deduction of $E$ from $\Gamma$ and $B$ satisfying the assumption of this theorem. In this deduction, $D_n$ is $E$. Let us show by hyperfinite induction that $\Gamma \vdash B \Rightarrow_s D_i$ for each $i \le n \in \mathbb{N}^#$. If $D_i$ is an axiom or belongs to $\Gamma$, then $\Gamma \vdash B \Rightarrow_s D_i$, since $D_i \Rightarrow_s (B \Rightarrow_s D_i)$ is an axiom. If $D_i$ is $B$, then $\Gamma \vdash B \Rightarrow_s D_i$, since, by Proposition 1, $\vdash B \Rightarrow_s B$. If there exist $j$ and $k$ less than $i$ such that $D_k$ is $\vdash D_j \Rightarrow_s D_i$, then, by inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$ and $\Gamma \vdash B \Rightarrow_s (D_j \Rightarrow_s D_i)$. Now, by axiom A2,

$\vdash B \Rightarrow_s (D_j \Rightarrow_s D_i) \Rightarrow_s ((B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s D_i))$. Hence, by MP twice, $\Gamma \vdash B \Rightarrow_s D_i$. Finally, suppose that there is some $j < i$ such that $D_i$ is $\forall x_k D_j$. By the inductive hypothesis, $\Gamma \vdash B \Rightarrow_s D_j$, and, by the hypothesis of the theorem, either $D_j$ does not depend upon $B$ or $x_k$ is not a free variable of $B$. If $D_j$ does not
depend upon B, then, by Proposition 2, $\Gamma \vdash D_j$ and, consequently, by
Gen, $\Gamma \vdash \forall x_k D_j$. Thus, $\Gamma \vdash D_i$. Now, by axiom A1, $\vdash D_i \Rightarrow_s (B \Rightarrow_s D_i)$.
So, $\Gamma \vdash B \Rightarrow_s D_i$ by MP. If, on the other hand, $x_k$ is not a free variable of B,
then, by axiom A5, $\vdash \forall x_k (B \Rightarrow_s D_j) \Rightarrow_s (B \Rightarrow_s \forall x_k D_j)$ Since $\Gamma \vdash B \Rightarrow_s D_j$,
we have, by Gen, $\Gamma \vdash \forall x_k (B \Rightarrow_s D_j)$, and so, by MP, $\Gamma \vdash B \Rightarrow_s \forall x_k D_j$
that is, $\Gamma \vdash B \Rightarrow_s D_i$. This completes the induction, and our proposition is
just the special case $i = n$.

## §2. Set theory $\mathbf{NC}_{\infty}^{\#}$.

Set theory $\mathbf{NC}_{\infty}^{\#}$ is formulated as a system of axioms based on bivalent hyper
infinitary logic $^{2}L_{\infty}^{\#}$ with restricted modus ponens rule [1],[18]. The language of set
theory $\mathbf{NC}_{\infty}^{\#}$ is a first-order hyper infinitary language $L_{\infty}^{\#}$ with equality =, which
includes a binary symbol $\in$. We write $x \neq y$ for $\neg (x = y)$ and $x \notin y$ for $\neg (x \in y)$.
Individual variables $x,y,z,\dots$, and $x^{\text{cl}},y^{\text{cl}},z^{\text{cl}},\dots$ of $L_{\infty}^{\#}$ will be understood as ranging
over classical sets. The unique existential quantifier $\exists!$ is introduced by writing, for any
formula $\varphi(x), \exists!x\varphi(x)$ as an abbreviation of the formula $\exists x[\varphi(x) \ \& \ \forall y(\varphi(y) \Rightarrow_s x = y)]$.
The language $L_{\infty}^{\#}$ will also contains the formation of terms of the form $\{x|\varphi(x)\}^{\text{NCL}}$, for
any formula $\varphi(x)$ containing the free variable $x$.

Such terms are called non-classical sets; we shall use upper case letters A, B, ..., and A<sup>NCL</sup>, B<sup>NCL</sup>, ... for such sets. For each non-classical set A = {x|φ(x)}<sup>NCL</sup> the formulas

$\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x)]$ and $\forall x[x \in A \Leftrightarrow_{s,w} \varphi(x,A)]$ is called the defining axioms for the non-classical set A.

**Remark 2.1.** Remind that in logic $^{2}L_{\infty}^{\#}$ with restricted modus ponens rule the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ does not always guarantee that

$$ \alpha, \alpha \Rightarrow \beta \vdash_{\mathrm{RMP}} \beta \qquad (2.1) $$

since for some $\alpha$ and $\beta$ possible

$$ \alpha, \alpha \Rightarrow \beta \vdash_{\mathrm{RMP}} \beta \qquad (2.2) $$

even if the statement $\alpha \wedge (\alpha \Rightarrow \beta)$ holds.

**Abbreviation 2.1.** We shall write for the sake of brevity instead (2.1) by

$$ \alpha \Rightarrow_s \beta \qquad (2.3) $$

and we shall write instead (2.2) by

$$ \alpha \Rightarrow_w \beta. \qquad (2.4) $$

**Remark 2.2.** Let A be an nonclassical set. Note that in set theory $\mathbf{NC}_{\infty}^{\#}$ the following true formula

$$ \exists A \forall x [x \in A \Leftrightarrow \varphi(x, A)] \qquad (2.5) $$

does not always guarantee that

$$ x \in A, x \in A \Rightarrow \varphi(x, A) \vdash_{\mathrm{RMP}} \varphi(x, A) \qquad (2.6) $$

even if $x \in A$ holds and (or)

$$ \varphi(x,A), \varphi(x,A) \Rightarrow x \in A \vdash_{\mathrm{RMP}} x \in A; \qquad (2.7) $$

even $\varphi(x,A)$ holds, since for nonclassical set A for some y possible
$$ y \in A, y \in A \Rightarrow \varphi(y,A) \not\vdash_{RMP} \varphi(y,A) \qquad (2.8) $$

and (or)

$$ \varphi(y,A), \varphi(y,A) \Rightarrow y \in A \not\vdash_{RMP} y \in A. \qquad (2.9) $$

**Remark 2.3.** Note that in this paper the formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x) \wedge x \in u] \qquad (2.10) $$

and more general formulas

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x,a) \wedge x \in u] \qquad (2.11) $$

is considered as the defining axioms for the classical set a.

**Remark 2.4.** Let a be a classical set. Note that in $NC^#_\infty$: (i) the following true formula

$$ \exists a \forall x [x \in a \Leftrightarrow \varphi(x,a) \wedge x \in u] \qquad (2.12) $$

always guarantee that

$$ x \in a, x \in a \Rightarrow \varphi(x,a) \vdash_{RMP} \varphi(x) \qquad (2.13) $$

if $x \in a$ holds and

$$ \varphi(x), \varphi(x) \Rightarrow x \in a \vdash_{RMP} x \in a; \qquad (2.14) $$

if $\varphi(x)$ holds;

In order to emphasize this fact mentioned above in Remark 2.1-2.3, we rewrite the defining axioms in general case for the nonclassical sets in the following

form

$$ \exists A \forall x \{[x \in A \Leftrightarrow_s \varphi(x,A)] \vee [x \in A \Leftrightarrow_w \varphi(x,A)]\} \qquad (2.15) $$

and similarly we rewrite the defining axioms in general case for the classical sets in the following form

$$ \exists a \forall x [x \in a \Leftrightarrow_s \varphi(x) \wedge (x \in u)]. \qquad (2.16) $$

**Abbreviation 2.2.** We write instead (2.15):

$$ \forall x \{[x \in A \Leftrightarrow_{s,w} \varphi(x,A)]\} \qquad (2.17) $$

**Definition 2.1.** (1) Let A be a nonclassical set defined by formula (2.17).

Assum that: (i) for some y statement $\varphi(y)$ and statement $\varphi(y) \Rightarrow y \in A$ holds and
(ii) $\varphi(y), \varphi(y) \Rightarrow y \in A \not\vdash_{RMP} y \in A, y \in A, y \in A \Rightarrow \varphi(y) \not\vdash_{RMP} \varphi(y)$.

Then we say that y is a weak member of non-classical set A and abbreviate $y \in_w A$.

**Abbreviation 2.3.** Let A be a nonclassical set defined by formula (2.17) We abbreviate $x \in_{s,w} A$ if the following statement $x \in_s A \vee x \in_w A$ holds, i.e.

$$ x \in_{s,w} A \leftrightarrow_{def} (x \in_s A \vee x \in_w A). \qquad (2.18) $$

**Definition 2.2.** (1) Two nonclassical sets A, B are defined to be equal and we write $A = B$ if $\forall x[x \in_{s,w} A \Leftrightarrow_s x \in_{s,w} B]$. (2) A is a subset of B, and we often write $A \subset_{s,w} B$, if $\forall x[x \in_{s,w} A \Rightarrow_s x \in_{s,w} B]$. (3) We also write **CL. Set**(a) for the formula $\exists u \forall x[x \in a \Leftrightarrow x \in u \wedge \varphi(x)]$. (4) We also write **NCL. Set**(A) for the formulas $\forall x[x \in_{s,w} A \Leftrightarrow_{s,v} \varphi(x)]$ and $\forall x[x \in_{s,v} A \Leftrightarrow_{s,v} \varphi(x,A)]$.

**Remark 2.5.** **CL. Set**(u) asserts that the set u is a classical set. For any classical set u,
it follows from the defining axiom for the classical set $u = \{x|x \in_s u \wedge \varphi(x)\}$ that
**CL. Set**($\{x|x \in_s u \wedge \varphi(x)\}$).

We shall identify {$x|x \in_s u$} with $u$, so that sets may be considered as (special sorts of)
nonclassical sets and we may introduce assertions such as $u \subset_s A, u \subseteq_s A$, etc.

**Abbreviation 2.4.** Let $\varphi(t)$ be a formula of $NC^\#_{\infty}$.

(i) $\forall x\varphi(x)$ and $\forall^{\text{CL}}x\varphi(x)$ abbreviates $\forall x(\text{CL. Set}(x) \Rightarrow \varphi(x))$

(ii) $\exists x\varphi(x)$ and $\exists^{\text{CL}}x\varphi(x)$ abbreviates $\exists x(\text{CL. Set}(x) \Rightarrow \varphi(x))$

(iii) $\forall X\varphi(X)$ and $\forall^{\text{NCL}}X\varphi(X)$ abbreviates $\forall X(\text{NCL. Set}(X) \Rightarrow \varphi(X))$

(iv) $\exists X\varphi(X)$ and $\exists^{\text{NCL}}X\varphi(X)$ abbreviates $\exists X(\text{NCL. Set}(X) \Rightarrow \varphi(X))$

**Remark 2.6.** If A is a nonclassical set, we write $\exists x \in A \varphi(x, A)$ for $\exists x[x \in A \wedge \varphi(x, A)]$ and $\forall x \in A \varphi(x, A)$ for $\forall x[x \in A \Rightarrow \varphi(x, A)]$.

We define now the following sets:

1. $\{u_1, u_2, \dots, u_n\} = \{x|x = u_1 \lor x = u_2 \lor \dots \lor x = u_n\}$. 2. $\{A_1, A_2, \dots, A_n\} =$
$= \{x|x = A_1 \lor x = A_2 \lor \dots \lor x = A_n\}$. 3. $\cup A = \{x|\exists y[y \in A \land x \in y]\}$.

4. $\cap A = \{x|\forall y[y \in A \Rightarrow x \in y]\}$. 5. $A \cup B = \{x|x \in A \lor x \in B\}$.

5. $A \cap B = \{x|x \in A \land x \in B\}$. 6. $A - B = \{x|x \in A \land x \notin B\}$. 7. $u^+ = u \cup \{u\}$.

8. $\mathbf{P}(A) = \{x|x \subseteq A\}$. 9. $\{x \in A | \varphi(x, A)\} = \{x|x \in A \land \varphi(x, A)\}$. 10. $\mathbf{V} = \{x|x = x\}$.

11. $\emptyset = \{x|x \neq x\}$.

The system $NC^\#_{\infty}$ of set theory is based on the following axioms:

**Extensionality1:** $\forall u \forall v[\forall x(x \in u \Leftrightarrow x \in v) \Rightarrow u = v]$

**Extensionality2:** $\forall A \forall B[\forall x(x \in A \Leftrightarrow_{s,w} x \in B) \Rightarrow A = B]$

**Universal Set:** NCL. Set(V)

**Empty Set:** CL. Set(Ø)

**Pairing1:** $\forall u \forall v \text{ CL. Set}(\{u, v\})$

**Pairing2:** $\forall A \forall B \text{ NCL. Set}(\{A, B\})$

**Union1:** $\forall u \text{ CL. Set}(\cup u)$

**Union2:** $\forall A \text{ NCL. Set}(\cup A)$

**Powerset1:** $\forall u \text{ CL. Set}(\mathbf{P}(u))$

**Powerset2:** $\forall A \text{ NCL. Set}(\mathbf{P}(A))$

**Infinity** $\exists a[\emptyset \in a \land \forall x \in a(x^+ \in a)]$

**Separation1** $\forall u_1 \forall u_2, \dots \forall u_n \forall a \exists \text{CL. Set}(\{x \in_s a | \varphi(x, u_1, u_2, \dots, u_n)\})$

**Separation2** $\forall u_1 \forall u_2, \dots \forall u_n \text{ NCL. Set}(\{x \in_{s,w} A | \varphi(x, A; u_1, u_2, \dots, u_n)\})$

**Comprehension1** $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x; u_1, u_2, \dots, u_n)]$

**Comprehension 2** $\forall u_1 \forall u_2, \dots \forall u_n \exists A \forall x[x \in_{s,w} A \Leftrightarrow_{s,w} \varphi(x, A; u_1, u_2, \dots, u_n)]$

**Comprehension 3** $\forall u_1 \forall u_2, \dots \forall u_n \exists a \forall x[x \in_s a \Leftrightarrow_s (a \subset u_1) \land \varphi(x, a; u_1, u_2, \dots, u_n)]$

In particular:

**Comprehension 3'** $\forall u \exists a \forall x[x \in_s a \Leftrightarrow_s (a \subset u) \land \varphi(x, a; u)]$

**Hyperinfinity:** see subsection 2.1.

**Remark 2.7.** Note that the axiom of hyper infinity follows from the schemata
Comprehension 3.

**Definition 2.3.** The ordered pair of two sets $u, v$ is defined as usual by

$$ \langle u, v \rangle = \{ \{u\}, \{u, v\} \}. \qquad (2.19) $$

**Definition 2.4.** We define the Cartesian product of two nonclassical sets A and B
as usual by

$$ A \times_{s,w} B = \{ \langle x, y \rangle \mid x \in_{s,w} A \land y \in_{s,w} B \} \qquad (2.20) $$
**Definition 2.5.** A binary relation between two nonclassical sets A, B is a subset $R \subseteq_{s,w} A \times_{s,w} B$. We also write $aR_{s,w}b$ for $< a, b > \in_{s,w} R$. The domain $\text{dom}(R)$ and the range $\text{ran}(R)$ of R are defined by

$$ \text{dom}(R) = \{x | \exists y (xR_{s,w}y)\}, \quad \text{ran}(R) = \{y : \exists x (xR_{s,w}y)\}. \qquad (2.21) $$

**Definition 2.6.** A relation $F_{s,w}$ is a function, or map, written $\text{Fun}(F_{s,w})$, if for each $a \in_{s,w} \text{dom}(F)$ there is a unique $b$ for which $aF_{s,w}b$. This unique $b$ is written $F(a)$ or $Fa$. We write $F_{s,w} : A \to B$ for the assertion that $F_{s,w}$ is a function with $\text{dom}(F_{s,w}) = A$ and $\text{ran}(F_{s,w}) = B$. In this case we write $a \mapsto F_{s,w}(a)$ for $F_{s,w}a$.

**Definition 2.7.** The identity map $1_A$ on A is the map $A \to A$ given by $a \mapsto a$. If $X \subseteq_{s,w} A$, the map $x \mapsto x : X \to A$ is called the insertion map of X into A.

**Definition 2.8.** If $F_{s,w} : A \to B$ and $X \subseteq_{s,w} A$, the restriction $F_{s,w}|_X$ of $F_{s,w}$ to X is the map $X \to A$ given by $x \mapsto F_{s,w}(x)$. If $Y \subseteq_{s,w} B$, the inverse image of Y under $F_{s,w}$ is the set

$$ F_{s,w}^{-1}[Y] = \{x \in_{s,w} A : F_{s,w}(x) \in_{s,w} Y\}. \qquad (2.22) $$

Given two functions $F_{s,w} : A \to B, G_{s,w} : B \to C$, we define the composite function

$G_{s,w} \circ F_{s,w} : A \to C$ to be the function $a \mapsto G_{s,w}(F_{s,w}(a))$. If $F_{s,w} : A \to A$, we write $F_{s,w}^2$ for $F_{s,w} \circ F_{s,w}, F_{s,w}^3$ for $F_{s,w} \circ F_{s,w} \circ F_{s,w}$ etc.

**Definition 2.9.** A function $F_{s,w} : A \to B$ is said to be monic if for all

$x, y \in_{s,w} A, F_{s,w}(x) = F_{s,w}(y)$ implies $x = y$, epi if for any $b \in_{s,w} B$ there is $a \in_{s,w} A$ for which $b = F_{s,w}(a)$, and bijective, or a bijection, if it is both monic and epi. It is easily shown that

$F_{s,w}$ is bijective if and only if $F_{s,w}$ has an inverse, that is, a map $G_{s,w} : B \to A$ such that $F_{s,w} \circ G_{s,w} = 1_B$ and $G_{s,w} \circ F_{s,w} = 1_A$.

**Definition 2.10.** Two sets X and Y are said to be equipollent, and we write $X \cong_{s,w} Y$, if there is a bijection between them.

**Definition 2.11.** Suppose we are given two sets $I, A$ and an epi map $F_{s,w} : I \to A$. Then $A = \{F_{s,w}(i) | i \in I\}$ and so, if, for each $i \in_{s,w} I$, we write $a_i$ for $F_{s,w}(i)$, then A can be presented in the form of an indexed set $\{a_i : i \in_{s,w} I\}$. If A is presented as an indexed set of sets $\{X_i | i \in_{s,w} I\}$, then we write $\bigcup_{i \in I} X_i$ and $\bigcap_{i \in I} X_i$ for $\bigcup A$ and $\bigcap A$, respectively.

**Definition 2.12.** The projection maps $\pi_1 : A \times_{s,w} B \to A$ and $\pi_2 : A \times_{s,w} B \to B$ are defined to be the maps $< a, b > \mapsto a$ and $< a, b > \mapsto b$ respectively.

**Definition 2.13.** For sets A, B, the exponential $B^A$ is defined to be the set of all functions from A to B.

## Axiom of nonregularity

Remind that a non-empty set $u$ is called regular iff $\forall x[x \neq \emptyset \to (\exists y \in x)(x \cap y = \emptyset)]$. Let's investigate what it says: suppose there were a non-empty $x$ such that $(\forall y \in x)(x \cap y \neq \emptyset)$. For any $z_1 \in x$ we would be able to get $z_2 \in z_1 \cap x$. Since $z_2 \in x$ we would be able to get $z_3 \in z_2 \cap x$. The process continues forever: $\dots \in z_{n+1} \in z_n \dots \in z_4 \in z_3 \in z_2 \in z_1 \in x$. Thus if we don't wish to rule out such an infinite regress we forced accept the following statement:

$$ \exists x[x \neq \emptyset \to (\forall y \in x)(x \cap y \neq \emptyset)]. \qquad (2.23) $$
## Axiom of hyperinfinity.

**Definition 2.14.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^{\infty}$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (2.24) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (2.25) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k+1]} = (a^{+[k]})^+$.

**Definition 2.15.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (2.26) $$

**Definition 2.16.** We say that a well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z|z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 2.17.** Assume $u \in \mathbb{N}^\#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^\#\mathbb{N}$.

### Axiom of hyperinfinity

There exists a set $\mathbb{N}^\#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^\#$,

(ii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $v$ such that $v < u$,

(iii) if $u \in \mathbb{N}^\#\mathbb{N}$ then there exists infinite (hypernatural) number $w$ such that for any $n \in \mathbb{N}$ : $u^{+[n]} < w$,

(iv) set $\mathbb{N}^\#\mathbb{N}$ is partially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

## Axiom of existence the nonclassical truth predicate

Let $A$ be a closed wff of $\mathbb{N}^\#$ ($\mathbb{N}^\#\mathbb{N}$-sentence). There is truth predicate $T^#[A]$ satisfies the following $T^#$-schema:

$$ T^#[A] \iff_{s,w} A. \qquad (2.27) $$

**Definition 2.18.** (i) We say that a $\mathbb{N}^\#$-sentence is a $s$-sentence (relative to $\vdash_{\mathrm{RMP}}$) if

$$ T^#[A] \iff_s A. \qquad (2.28) $$

(ii) We say that a $\mathbb{N}^\#$-sentence is a $w$-sentence (relative to $\vdash_{\mathrm{RMP}}$) if

$$ T^#[A] \iff_w A. \qquad (2.29) $$

**Remark 2.8.** Remind that classical truth predicate $T[A]$ unrestrictedly satisfies the following $T$-schema [24]:
$$ T[A] \Leftrightarrow A, \qquad (2.30) $$

i.e., the sentence $A \Leftrightarrow T[A]$ is true for every sentence $A$ of language $L$, where $T[A]$ stands for "the sentence (denoted by) $A$ is true". Unfortunately T-schema incorrect by well known Curry's paradox.

Assume, too, that we have the principle called Assertion (also known as pseudo modus ponens): $(A \land (A \Rightarrow B)) \Rightarrow B$. By diagonalization, self-reference we can get a sentence $C$ such that $C \Leftrightarrow (T[C] \Rightarrow F)$ where $F$ is anything you like. (For effect, though, make $F$ something obviously false, e.g. $F \equiv \bot = 0 = 1$) By an instance of the T-schema: $T[C] \Leftrightarrow C$ we immediately get: $T[C] \Leftrightarrow (T[C] \Rightarrow F)$. Again, using the same

instance of the T-schema, we can substitute $C[T, F]$ for $\mathcal{T}[C]$ in the above to get (1).

(1) $\vdash C[T, F] \Leftrightarrow (C[T, F] \Rightarrow F)$ [by T-schema and substitution]

(2) $\vdash (C[T, F] \land (C[T, F] \Rightarrow F)) \Rightarrow F$ [by assertion]

(3) $\vdash (C[T, F] \land C[T, F]) \Rightarrow F$ [by substitution, from (2)]

(4) $\vdash C[T, F] \Rightarrow F$ [by equivalence of $C$ and $C \land C$, from (3)]

(5) $\vdash C[T, F]$ [by unrestricted Modus Ponens, from (1) and (4)]

(6) $\vdash F$ [by unrestricted Modus Ponens, from (4) and (5)]

Letting $F$ be anything entailing triviality Curry's paradox quickly 'shows' that the world is trivial.

## §3. Nonconservative extension of the model theoretical NSA based on bivalent hyper Infinitary first-order logic $^{2}L_{\infty}^{\#}$ with restricted canonical rules of conclusion.

Extending the classical real numbers $\mathbb{R}$ to include infinite and infinitesimal quantities originally enabled D. Laugwitz [1] to view the delta distribution $\delta(x)$ as a nonstandard point function. Independently A. Robinson [2] demonstrated that distributions could be viewed as generalized polynomials. Luxemburg [3] and Sloan [4] presented an alternate representative of distributions as internal functions within the context of canonical Robinson's theory of nonstandard analysis. For further information on classical nonstandard real analysis, we refer to [8]-[11].

**Abbreviation 3.1.** In this paper we adopt the following notations. For a standard set $E$ we often write $E_{st}$. For a set $E_{st}$ let $^{\sigma}E_{st}$ be a set $^{\sigma}E_{st} = \{*x|x \in E_{st}\}$. We identify $z$ with $^{\sigma}z$ i.e., $z \equiv^{\sigma} z$ for all $z \in \mathbb{C}$. Hence, $^{\sigma}E_{st} = E_{st}$ if $E \subseteq \mathbb{C}$, e.g., $^{\sigma}\mathbb{C} = \mathbb{C}$, $^{\sigma}\mathbb{R} = \mathbb{R}$, $^{\sigma}P = P$, $^{\sigma}L_{\uparrow}^{+} = L_{\uparrow}^{+}$, etc. Let $^{*}\mathbb{R}_{\approx}$, $^{*}\mathbb{R}_{\approx,+}$, $^{*}\mathbb{R}_{\text{fin}}$, $^{*}\mathbb{R}_{\infty}$, and $^{*}\mathbb{N}_{\infty}$ denote the sets of infinitesimal hyper-real numbers, positive infinitesimal hyper-real numbers, finite hyper-real numbers, infinite hyper-real numbers and infinite hyper natural numbers, respectively. Note that $^{*}\mathbb{R}_{\text{fin}} = *\mathbb{R}/*\mathbb{R}_{\infty}$, $^{*}\mathbb{C} = *\mathbb{R} + i*\mathbb{R}$, $^{*}\mathbb{C}_{\text{fin}} = *\mathbb{R}_{\text{fin}} + i*\mathbb{R}_{\text{fin}}$.

Remind that Robinson nonstandard analysis (RNA) many developed using set-theoretical objects called superstructures [8]-[11]. A superstructure $V(S)$ over a set $S$ is defined in the following way

$$ V_0(S) = S, V_{n+1}(S) = V_n(S) \cup (P(V_n(S)), V(S) = \bigcup_{n \in \mathbb{N}} V_n(S)). \qquad (3.1) $$

Superstructures of the empty set consist of sets of infinite rank in the cumulative hierarchy and therefore do not satisfy the in...nity axiom. Making $S = \mathbb{R}$ will suffice for
virtually any construction necessary in analysis.

Bounded formulas are formulas where all quantifiers occur in the form

$$ \forall x(x \in y \Rightarrow \dots), \exists x(x \in y \Rightarrow \dots). \qquad (3.2) $$

A nonstandard embedding is a mapping

$$*: \mathbf{V}(X) \to \mathbf{V}(Y)$$

from a superstructure $\mathbf{V}(X)$ called the standard universum, into another superstructure

$\mathbf{V}(Y)$, called nonstandard universum, satisfying the following postulates:

**1.** $Y = *X$

**2. Transfer Principle.** For every bounded formula $\Phi(x_1, \dots, x_n)$ and elements $a_1, \dots, a_n \in \mathbf{V}(X)$, the property $\Phi$ is true for $a_1, \dots, a_n$ in the standard universum if and only if it is true for $*a_1, \dots, *a_n$ in the nonstandard universum:

$$ \langle \mathbf{V}(X), \in \rangle \models \Phi(a_1, \dots, a_n) \Leftrightarrow \langle \mathbf{V}(Y), \in \rangle \models \Phi(*a_1, \dots, *a_n). $$

**3. Non-triviality.** For every infinite set $A$ in the standard universum, the set $\{*a | a \in A\}$ is a proper subset of $*A$.

**Definition 3.1.** [10]. A set $x$ is **internal** if and only if $x$ is an element of $*A$ for some element $A$ of $\mathbf{V}(\mathbb{R})$. Let $X$ be a set with $A = \{A_i\}_{i \in I}$ a family of subsets of $X$. Then the collection $A$ has the infinite intersection property, if any infinite subcollection $J \subset I$ has non-empty intersection. Nonstandard universum is $\kappa$-saturated if whenever $\{A_i\}_{i \in I}$ is a collection of internal sets with the infinite intersection property and the cardinality of $I$ is less than or equal to $\kappa$, $\bigcap_{i \in I} A_i \neq \emptyset$.

**Remark 3.1.** Remind that: (i) for each standard universum $U = \mathbf{V}(X)$ there exists canonical language $\mathcal{L} = \mathcal{L}_U$, (ii) for each nonstandard universum $W = \mathbf{V}(Y)$ there exists corresponding canonical nonstandard language $*\mathcal{L} = \mathcal{L}_W$ [10].

**3\*. The restricted rules of conclusion.**

If $W \models A$ then $\neg A \not\models B$, where $B \in \mathcal{L} \wedge B \in *\mathcal{L}$.

Thus if $A$ holds in $W$ we cannot obtain from $\neg A$ any formula $B$ whatsoever.

**Remark 3.2.** We write $* \models A$ instead $W \models A$.

In this paper we apply the following hyper inductive definitions of a sets [18]

$$ \exists S \forall \beta (\beta \in *\mathbb{N}) \left[ \beta \in S \Leftrightarrow_s \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right]. $$

**Definition 3.2.** [18]. A set $S \subset *\mathbb{N}$ is a **hyper inductive** if the following statement holds

$$ \bigwedge_{\alpha \in *\mathbb{N}} (\alpha \in S \Rightarrow_s \alpha^+ \in S), \qquad (3.3) $$

where $\alpha^+ \triangleq \alpha + 1$. Obviously a set $*\mathbb{N}$ is a hyper inductive. As we see later there is just one hyper inductive subset of $*\mathbb{N}$, namely $*\text{Nitself}$.

We extend up Robinson nonstandard analysis (RNA) by adding the following postulate:

**4. Any hyper inductive set $S$ is internal.**

**Remark 3.3.** The statement 4 is not provable in ZFC but provable in set theory $\mathbb{N}C^\#_\infty$, see [2]-[3]. Thus postulates 1-4 gives an nonconservative extension of RNA and we denote such extension by **NERNA**.

**Remark 3.4.** Note that NERNA of course based on the same hyper infinitary logic with
Restricted Modus Ponens Rule as set theory $NC_{\infty}^{\#}$ [1]-[3].

Remind that in RNA the following induction principle holds.

**Theorem 3.1.**[6]. Assume that $S \subset *\mathbb{N}$ is internal set, then

$$ (1 \in S) \wedge \forall x[x \in S \Rightarrow x + 1] \Rightarrow S = *\mathbb{N}. \qquad (3.4) $$

In NERNA Theorem 1.1 also holds.

**Remark 3.5.** It follows from postulate 4 and Theorem 1.1 that any hyper inductive set $S$ is equivalent to $*\mathbb{N} : S = *\mathbb{N}$.

**Remark 3.6.** Note that the following statements are provable in $NC_{\infty}^{\#}$ [2]-[3]:

**5 Axiom of $\omega$-induction**

$$ \forall S(S \subset_s \mathbb{N}) \left\{ \forall \beta(\beta \in_s \mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S) \right] \Rightarrow_s S = \mathbb{N} \right\}. \qquad (3.5) $$

**6 Axiom of hyper infinite induction**

$$ \forall S(S \subset *\mathbb{N}) \left\{ \forall \beta(\beta \in *\mathbb{N}) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow_s \alpha^+ \in S) \right] \Rightarrow_s S = *\mathbb{N} \right\}. \qquad (3.6) $$

Thus postulate 5 of the theory NERNA is provable in $NC_{\infty}^{\#}$.

**Rules of conclusion**

(1) Restricted Modus Ponens Rule (denoted by $\vdash_{RMP}$) the same as in set theory $NC_{\infty}^{\#}$.

(2) Restricted Modus Tollens Rule (denoted by $\vdash_{RMT}$) the same as in set theory $NC_{\infty}^{\#}$.

(3) MRR1 (1. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n}(\bar{n} \in *\mathbb{N}\mathbb{N}) \wedge \mathbf{V}(Y) \models \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg\varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg\varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

(4) MRR2 (2. Main Restricted rule of conclusion)

Let $\varphi(x)$ be a wff with one free variable $x$ and such that $\exists \bar{n}(\bar{n} \in *\mathbb{N}) \wedge \mathbf{V}(Y) \models \varphi(\bar{n})$, then for all $n \ge \bar{n} : \neg\varphi(n) \vdash_{RMP} B$, i.e., if statement $\varphi(\bar{n})$ holds in $\mathbf{V}(Y)$ we cannot obtain from $\neg\varphi(n)$, with $n \ge \bar{n}$ any formula $B$ whatsoever.

**Remark 3.5.** The MRR1,2 is necessarily in natural way, since by assumption $\neg\varphi(n)$ one obtains directly the apparent contradiction $\varphi(n) \wedge \neg\varphi(n)$ from which by unrestricted modus ponens rule (UMPR) one obtains $\varphi(n) \wedge \neg\varphi(n) \vdash_{UMPR} B$.

**Example 3.1.** Remind the proof of the following statement:

**Theorem 3.2.** The structure $(\mathbb{N}, <)$ is a well-ordered set.

**Proof.** Let $X$ be a nonempty subset of $\mathbb{N}$. Suppose $X$ does not have a $<$-least element. Then consider the set $\mathbb{N}\backslash X$.

Case (1) $\mathbb{N}\backslash X = \emptyset$. Then $X = \mathbb{N}$ and so $0$ is a $<$-least element. Contradiction.

Case (2) $\mathbb{N}\backslash X \neq \emptyset$. Then $1 \in \mathbb{N}\backslash X$ otherwise $1$ is a $<$-least element. Contradiction.

Case (3) $\mathbb{N}\backslash X \neq \emptyset$. Assume now that there exists an $n \in \mathbb{N}\backslash X$ such that $n \neq 1$.

Since we have supposed that $X$ does not have a least element, thus $n + 1 \notin X$.

Thus we see that for all $n : n \in \mathbb{N}\backslash X$ implies that $n + 1 \in \mathbb{N}\backslash X$. We can conclude by induction that $n \in \mathbb{N}\backslash X$ for all $n \in \mathbb{N}$. Thus $\mathbb{N}\backslash X = \mathbb{N}$ implies $X = \emptyset$.

This is a contradiction to $X$ being a nonempty subset of $\mathbb{N}$.

**Remark 3.6.** (i) The proof of the Theorem 3.2 is an example proof by a contradiction.

Remind that a mathematical proof employing proof by contradiction usually proceeds
as follows:

1. The proposition to be proved is P.

2. We assume P to be false, i.e., we assume $\models \neg P$.

3. It is then shown that $\neg P$ implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, $Q$ and $\neg Q$, and appealing to the law of noncontradiction.

4. Since assuming P to be false leads to a contradiction, it is concluded that P is in fact true.

(ii) The statement of the Theorem 3.2 obviously is unprovable by a contradiction under MRR2. Note that in the Case (3) there is an $\bar{n} \neq 1, \bar{n} \in X$ and $\bar{n} \notin N \setminus X$. Thus induction hypothesis $\models \bar{n} \in N \setminus X$ is not holds since $\bar{n} \notin N \setminus X \land \bar{n} \in N \setminus X$ is a contradiction and by MRR2

$$ \bar{n} \in N \setminus X \vdash_R \mathrm{RMP} \ \bar{n} + 1 \in N \setminus X. $$

(iii) Note that proof of the Theorem 3.2 mentioned above completely abnormal in fact even in point view of classical proof theory, since basic assumption $\bar{n} \notin N \setminus X$ which is employed in proof by contradiction, contradicts with induction hypothesis $\models \bar{n} \in N \setminus X$.

**Example 3.2.** (i) We set now $X_1 = *N \setminus N$, thus $*N \setminus X_1 = N$. In contrast with a set $X$ mentioned in Example 3.1, the assumption $n \in *N \setminus X_1$ implies that $n + 1 \in *N \setminus X_1$ if and only if $n$ is finite, since for any infinite $n \in *N \setminus N$ the assumption $n \in *N \setminus X_1$ contradicts with a true statement $V(Y) \models n \notin *N \setminus X_1 = N$ and therefore in accordance with MRR we cannot obtain for any infinite $n$ from formula $n \in *N \setminus X_1$ any formula $B$ whatsoever.

**Remark 3.7.** Notice in order to prove an statement $G = \forall n(n \in *N)P(n)$ by induction one needs to proof that: $P(n) \vdash_R \mathrm{RMP} P(n+1)$, i.e. by assuming that $P(n)$ is true and then by RMP proving $P(n+1)$. Thus:

(i) any proof by hyperinfinite induction bused on additional assumption that

$$ \vdash_R \mathrm{RMP} \exists \bar{n} (\bar{n} \in_s *N)[\neg P(\bar{n})]. \qquad (3.7) $$

(ii) any proof by $\omega$-induction bused on additional assumption that

$$ \vdash_R \mathrm{RMP} \exists \bar{n} (\bar{n} \in_s N)[\neg P(\bar{n})]. \qquad (3.8) $$

**Definition 3.3.** $\kappa$ is a natural number if $\kappa \in_S X$ for every set $X$ such that $0 \in_S X$ and, for any $\lambda$, if $\lambda \in_S X$ then $\lambda + 1 \in_S X$, i.e. $\lambda \in_S X \vdash_R \mathrm{RMP} \ \lambda + 1 \in_S X$.

**We remind now some basic theorem and definitions related to classical naturals.**

**Definition 3.4.** [20]. $\kappa$ is a natural number if $\kappa$ belongs to every set $X$ such that $0 \in X$ and, for any $\lambda$, if $\lambda \in X$ then $\lambda + 1 \in X$.

(As usual, $j,k,...,n$ will denote natural numbers.)

**Remark 3.8.** [20]. If the set of all natural numbers exists, we call it $N$. But it is not necessary for us to assume now that $N$ exists. The assumption that $N$ exists is a form of what is called the Axiom of Infinity.

**Proposition 3.1.** [20] For any $\kappa$, $\{\lambda|\lambda \le \kappa\}$ exists.

**Proof.** Let $\bar{A} = \kappa$. The desired set is $\{\bar{B}|B \in_S P(A)\}$, which exists by the Axiom of Replacement.

**Theorem 3.3.** [20].
(a) 0 is a natural number, $0 \le k$

(b) If $k$ is a natural number so is $k+1$, if $k < \lambda$ then $k+1 \le \lambda$.

(c) (Induction) Suppose that $P(0)$ ('P holds for 0'); and that, for any natural number $n$, $P(n) \Rightarrow P(n+1)$ holds. Then for every $n$, $P(n)$ holds.

**Proof** [20] (a) and (b) are very easy. For (c), suppose that the whole hypothesis of (c) holds, but that, for some particular $\bar{n}, P(\bar{n})$ fails, i.e. $\neg P(\bar{n})$ holds. Put

$$X = \{m < \bar{n} | P(m)\} \qquad (3.9)$$

$X$ exists since $X = \{\lambda|\lambda < \eta$ and $\lambda$ is a natural number and $P(\lambda)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $\lambda + 1 \in X$ whenever $\lambda \in X$ - as then $X$ is 'an $X$' as in Definition 3.3, so, by Definition 3.3, the natural number $\bar{n} \in X$, and so $P(\bar{n})$ holds, a contradiction. Suppose then that $\lambda \in X$, so that $\lambda \le \bar{n}, \lambda$ is a natural number, and $P(\lambda)$. By our hypothesis (in (c)), $P(\lambda + 1)$. By (b), $\lambda + 1$ is a natural number. Also, $\lambda < \bar{n}$, as $P(\bar{n})$ fails. Hence $\lambda + 1 \le n$, by Theorem 3.3 (b). So $\lambda + 1 \in X$, as desired.

**Remark 3.9.** Note that proof of the Theorem 3.3 mentioned above completely abnormal since definition (3.9) incorrect. Correct definition reads

$$X = \{m < \bar{n} | \mathbf{T}[P(m)] \wedge P(m)\} \qquad (3.10)$$

where $\mathbf{T}[A]$ is a truth predicate such that for any well formed closed formula $A$ of ZFC [24]

$$\mathbf{T}[A] \Leftrightarrow A. \qquad (3.11)$$

However as well known such truth predicate is not exists by Curry's paradox. Thus a set $X$ is not exists in general case.

**Definition 3.5.** An element $x$ is said to be a first element of the linearly ordered set $A$ (with respect to the relation $R$) if $xRy$ for all $y \in A$. On the other hand, if $yRx$ for all $y$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Theorem 3.4.** [20]-[22]. In a finite non-empty subset $X$ of a linearly ordered set $A$ there is a first element and a last element of $X$.

**Proof.** The proof is by induction on the number of elements of $X$. If $X$ has only one element, then the theorem is obvious. Suppose that the theorem holds for subsets with $n$ elements. Let $X = Y \cup \{a\}$ where $a \notin Y$ and $Y$ has $n$ elements. Let $b_1$ be the first and $b_2$ the last element of $Y$. Since $A$ is linearly ordered, either a precedes $b_1$ or $b_1$ precedes $a$. That element which precedes the other is clearly the first element of $Y$. Similarly we show that one of the elements $a$ and $b_2$ is the last element of $X$.

**Corollary 3.1.** Every finite subset $A$ of $\mathbb{N}$ has a first element and also a last element.

**Proof.** From Theorem 3.4 by definitions.

**Theorem 3.5.** [20]. (a) **(The least element principle)**. If for some $\bar{n}, P(\bar{n})$, then there is a minimal (which is here the same as minimum) $n$ such that $P(n)$.

(b) **(Course-of-values induction)**. If, for any $n$, if $Q(m)$ holds for all $m < n$, then $Q(n)$; then, for all $n, Q(n)$.

**Proof of (a).** Suppose $P(\bar{n})$. If $(P(m)$ for no $m < \bar{n}$, then $\bar{n}$ is minimal as desired. Otherwise $\{k \in W(\bar{n}) | P(k)\} (W(\bar{n}) = \{m|m < n\})$ is non-empty, and so, being finite,
has a least element $m$, by Corollary3.1. It is easy to see that $m$ is the least number with the property $P$, as desired.

**Proof** of (b). Assume the hypothesis of (b) holds and that, for some $n$, $Q(n)$. fails. By (a) let $k$ be the least such $n$. Thus $Q(m)$ holds for all $m < k$, so by our hypothesis, $Q(k)$ holds, a contradiction.

**Theorem 3.6.** (Induction) Let $P(x)$ be wff of $\mathbf{NC}_{\infty}^{\#}$ with a free variable $x$. Suppose that

$$ \mathbf{T}^{\#}[P(0)] \wedge \mathbf{T}^{\#}[P(0)] \qquad (3.12) $$

($P$ holds for $0$); $\mathbf{T}^{\#}[P(0)] \Leftrightarrow_S P(0)$, and that, for any natural number $n$,

$$ P(n) \Rightarrow_S P(n+1) \qquad (3.13) $$

and for every $n$,

$$ \mathbf{T}^{\#}[P(n)] \Leftrightarrow_S P(n). \qquad (3.14) $$

and

$$ \mathbf{T}^{\#}[P(n)] \Leftrightarrow_S P(n) \not\vdash_{\mathrm{RMT}} \mathbf{T}^{\#}[P(n)] \wedge P(n). \qquad (3.15) $$

Then for every $n$, $P(n)$ holds, i.e. $\forall n\{\mathbf{T}^{\#}[P(n)] \wedge P(n)\}$.

**Proof.** Suppose that the whole hypothesis mentioned above holds, but that, for some particular $\bar{n}, P(\bar{n})$ fails, i.e. $-P(\bar{n})$ holds. Put

$$ X = \{m < \bar{n} | \mathbf{T}^{\#}[P(m)] \wedge P(m)\} \qquad (3.16) $$

$X$ exists since $X = \{\lambda|\lambda < \eta$ and $\lambda$ is a natural number and $\mathbf{T}^{\#}[P(m)] \wedge P(\lambda)\}$, which exists by Proposition 3.1 and the Separation Axiom. Obviously, $0 \in X$ by Theorem 3.3 (a). It will be enough to show that: $\lambda + 1 \in_S X$ whenever $\lambda \in_S X$ - as then $X$ is 'an $X'$ as in Definition 3.3, so, by Definition 3.3, the natural number $\bar{n} \in_S X$,

and so $P(\bar{n})$ holds, a contradiction. Suppose then that $\lambda \in_S X$, so that $\lambda \le \bar{n}, \lambda$ is a natural number, and $P(\lambda)$. By our hypothesis (in (3.13)), $P(\lambda + 1)$. By (b), $\lambda + 1$ is a natural number. Also, $\lambda < \bar{n}$, as $P(\bar{n})$ fails. Hence $\lambda + 1 \le n$, by Theorem 3.3 (b). So $\lambda + 1 \in_S X$, as desired.

**Theorem 3.7.** [23] Any finite nonempty subset $X$ of $\mathbb{N}$ has minimal and maximal members.

**Proof** [23]. Let $X_n$ consist of $x_1, \dots, x_n$. Define $m_1 = x_1$ and $m_k$ as $x_k$ if $x_k < m_{k-1}$ and $m_{k-1}$ otherwise. Then $m_n$ will be minimal. Similarly, $X$ has a maximal element.

**Remark 3.7.** This proof in fact based on assumption (the induction hypothesis) that the theorem holds for $X_{k-1}$ consist of $x_1, \dots, x_{k-1}$, i.e. $m_{k-1} = \min\{x_1, \dots, x_{k-1}\}$, then it follows $m_{k-1} = \min\{x_1, \dots, x_{k-1}\} \Rightarrow m_k = \min\{x_1, \dots, x_k\}$ and by induction we conclude that for all $n \in \mathbb{N}$, $m_n = \min\{x_1, \dots, x_n\}$.

**Definition 3.6.** An element $x$ is said to be a first element of the linearly s-ordered set $A$ (with respect to the s-relation $R$) if $xRy$ for all $y \in_S A$. On the other hand, if $yRx$ for all $y \in_S A$, then $x$ is said to be a last element of $A$ (with respect to $R$). Generally speaking, not every set has a first or last element; but if such an element exists, then it is uniquely determined.

**Abbreviation 3.2** Let $X_n(A), \bar{X}_n(A) = n$ be a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n$. We shall abbreviated: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered
set A such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$.
Under assumption

$$ \Vdash_{\mathrm{RMP}} \exists m (m \in s \ \mathbb{N}) \exists X_m(A) [\neg \hat{X}_m(A)]. \qquad (3.9) $$

by axiom of ω-induction we obtain

$$ \forall X_n(A) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(A) \Rightarrow_s \hat{X}_{n+1}(A)) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(A)]. \qquad (3.10) $$

In particular for $A = \mathbb{N}$ under assumption

$$ \Vdash_{\mathrm{RMP}} \exists m (m \in s \ \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (3.11) $$

by axiom of ω-induction we obtain

$$ \forall X_n(\mathbb{N}) \left[ \bigwedge_{n \in \mathbb{N}} (\hat{X}_n(\mathbb{N}) \Rightarrow_s \hat{X}_{n+1}(\mathbb{N})) \right] \Rightarrow_s \forall n \forall X_n(A) [\hat{X}_n(A)]. \qquad (3.12) $$

**7 Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**

$$ \exists m (m \in s \ \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (3.13) $$

## §4. Internal Set Theory IST.

The axiomatics IST (Internal Set Theory) was presented in 1977 [19] and in a sense formulates within first-order language the behaviour of standard and internal sets of a nonstandard model of ZFC. This were done by adding the unary standardness predicate "st" to the language of ZFC as well as adding to the axioms of ZFC three new axiom schemes involving the predicate "st": **Idealization**, **Standardization** and **Transfer**.

**Remark 4.1.** Formulas which do not use the predicate st are called **internal formulas** (or $\in$-formulas) and formulas that use this new predicate are called **external formulas** (or st-$\in$-formulas). A formula $\varphi$ is standard if only standard constants occur in $\varphi$.

**Abbreviation 4.1.** We denote a set of the all naturals by $\mathbb{N}^\#\text{and a set of the all finite naturals by } \mathbb{N}$.

**Abbreviation 4.2.** We write $\mathbf{fin}(x)$ meaning 'x is finite'. Let $\varphi(x)$ be a st-$\in$-formula:

1. $\forall^{\mathbf{st}} x \varphi(x)$ abbreviates $\forall x (\mathbf{st}(x) \Rightarrow \varphi(x))$.

2. $\exists^{\mathbf{st}} x \varphi(x)$ abbreviates $\exists x (\mathbf{st}(x) \wedge \varphi(x))$.

3. $\forall^{\mathbf{fin}} x \varphi(x)$ abbreviates $\forall x (\mathbf{fin}(x)) \Rightarrow \varphi(x)$.

4. $\exists^{\mathbf{fin}} x \varphi(x)$ abbreviates $\exists x (\mathbf{fin}(x) \wedge \varphi(x))$.

The fundamental axioms of IST :

### (I) Idealization

$$ \forall^{\mathbf{stfin}} F \exists y \forall x \in F [R(x, y) \iff \exists b \forall^{\mathbf{st}} x R(x, b)] \qquad (4.1) $$

for any internal relation $R$.

**Remark 4.2.** The idealization axiom obviously states that saying that for any fixed finite set $F$ there is a $y$ such that $R(x, y)$ holds for all $x \in F$ is the same as saying that there is a $b$ such that for all fixed $x$ the relation $R(x, b)$ holds.

### (II) Standardization

$$ \forall^{\mathbf{st}} A \exists^{\mathbf{st}} B \forall^{\mathbf{st}} x (x \in B \Leftrightarrow x \in A \wedge \varphi(x)) \qquad (4.2) $$

for every st-$\in$-formula $\varphi$ with arbitrary (internal) parameters.
(III) Transfer

$$
\forall^{\text{st}} y_1, \dots, y_n \forall^{\text{st}} x [\varphi(x, y_1, \dots, y_n)] \Rightarrow \forall x \varphi(x, y_1, \dots, y_n) \quad (4.3)
$$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Remark 5.3.** An important consequence of (I) is the principle of **External Induction**,
which states that for any (external or internal) formula $\varphi$, one has

$$
\varphi(0) \wedge [\forall^{\text{st}} n (\varphi(n) \Rightarrow \varphi(n+1))] \Rightarrow \forall^{\text{st}} n \varphi(n). \quad (4.4)
$$

Boundedness

$$
\forall x \exists^{\text{st}} y (x \in y) \qquad (4.5)
$$

and since (2.5) contradicts idealization the following (bounded) form is taken instead:

(IV) Bounded Idealization

For every $\epsilon$-formula $R$ :

$$
\forall^{\text{st}} Y[\forall^{\text{stfin}} F \exists y \in Y (\forall x \in FR(x,y) \Leftrightarrow \exists b (b \in Y) \forall^{\text{st}} x R(x,b))]. \quad (4.6)
$$

This gives a subsystem BST, which corresponds to the bounded sets of IST.

§5.Internal Set Theory IST#

The axiomatics **IST**\# formulates within infinitary first-order language the behaviour
of standard and internal sets of a nonstandard model of $\mathbf{NC}_{\infty}^{\#}$. This done by adding
the unary standardness predicate "st" to the language of $\mathbf{NC}_{\infty}^{\#}$ as well as adding to
the axioms of $\mathbf{NC}_{\infty}^{\#}$ three new axiom schemes involving the predicate "st":

Idealization, Standardization, Transfer and Axiom of internal hyper infinite
induction.

**Remark 5.1.** Formulas which do not use the predicate st are called *internal* formulas (or *∈<sub>sw</sub>* -formulas) and formulas that use this new predicate are called *external* formulas (or st-*∈<sub>sw</sub>* -formulas). A formula φ is standard if only standard constants occur in φ.

**Abbreviation 5.1.** We write **fin**(x) meaning 'x is finite'. Let φ(x) be a **st**- *∈<sub>sw</sub>* -formula:

1. $\forall_s^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_s \varphi(x))$.

2. $\forall_{s,w}^{\text{st}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \Rightarrow_{s,w} \varphi(x))$.

3. $\exists^{\text{st}} x \varphi(x)$ abbreviates $\exists x (\mathbf{st}(x) \wedge \varphi(x))$.

4. $\forall_s^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\mathbf{fin}(x)) \Rightarrow_s \varphi(x)$.

5. $\forall_{s,w}^{\text{fin}} x \varphi(x)$ abbreviates $\forall x (\mathbf{fin}(x)) \Rightarrow_{s,w} \varphi(x)$.

6. $\exists^{\text{fin}} x \varphi(x)$ abbreviates $\exists x (\mathbf{fin}(x) \wedge \varphi(x))$.

7. $\forall_s^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x)) \Rightarrow_s \varphi(x)$.

8. $\forall_{s,w}^{\text{stfin}} x \varphi(x)$ abbreviates $\forall x (\text{st}(x) \wedge \text{fin}(x)) \Rightarrow_{s,w} \varphi(x)$.

9. $\exists^{\text{stfin}} x \varphi(x)$ abbreviates $\exists x (\text{st}(x) \wedge \text{fin}(x) \wedge \varphi(x))$.

The fundamental axioms of IST\# :

(I) Idealization for classical sets

$$
\forall_s^{\text{stfin}} F^{\text{CL}} \exists y^{\text{CL}} \forall x^{\text{CL}} \in_s F[R^{\text{CL}}(x,y) \Leftrightarrow_s \exists b^{\text{CL}} \forall_s^{\text{st}} x R^{\text{CL}}(x,b)] \quad (5.1)
$$

for any internal classical relation $R^{\text{CL}}(x, y)$.

**Remark 5.2.** The idealization axiom obviously states that saying that for any fixed classical finite set *F* there is a classical *y* such that *R*<sup>CL</sup>(*x*, *y*) holds for all classical *x* ∈ *S* *F* is the same as saying that there is a classical *b* such that for all fixed classical
x the classical relation $R^{\text{CL}}(x, b)$ holds.

(II) Standardization for classical sets

$$ \forall^{\text{st}} A^{\text{CL}} \exists^{\text{st}} B^{\text{CL}} \forall^{\text{st}} x^{\text{CL}} (x \in B \Leftrightarrow_s x \in A \land \varphi(x)) \qquad (5.2) $$

for every st-epsilon formula $\varphi$ with arbitrary (internal) parameters.

(III) Transfer for classical sets

$$ \forall^{\text{st}} y_1^{\text{CL}}, \dots, y_n^{\text{CL}} \forall^{\text{st}} x^{\text{CL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_s \forall x^{\text{CL}} \varphi(x, y_1, \dots, y_n) \qquad (5.3) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Boundedness**

$$ \forall x^{\text{CL}} \exists^{\text{st}} y^{\text{CL}} (x \in_s y) \qquad (5.4) $$

and since (5.4) contradicts idealization the following (bounded) form is taken instead:

(IV) Bounded Idealization for classical sets

For every $\epsilon$-formula $R$:

$$ \forall^{\text{st}} Y^{\text{CL}} [\forall^{\text{stfin}} F^{\text{CL}} \exists^{\text{y}} y^{\text{CL}} \in Y (\forall x^{\text{CL}} (x \in F) R(x, y) \Leftrightarrow_s \exists b^{\text{CL}} (b \in Y) \forall^{\text{st}} x R(x, b))] \qquad (5.5) $$

(V) Idealization for nonclassical sets

$$ \forall^{\text{stfin}}_{s,w} F^{\text{NCL}} \exists^{\text{y}} y^{\text{NCL}} \forall x^{\text{NCL}} \in_{s,w} F [R^{\text{NCL}}(x, y) \Leftrightarrow_{s,w} \exists b^{\text{NCL}} \forall^{\text{st}}_{s,w} x R^{\text{NCL}}(x, b)] \qquad (5.6) $$

for any internal nonclassical relation $R^{\text{NCL}}(x, y)$.

**Remark 5.3.** The idealization axiom obviously states that saying that for any fixed nonclassical finite set $F$ there is a classical $y$ such that $R^{\text{NCL}}(x, y)$ holds for all classical $x \in_s F$ is the same as saying that there is a classical $b$ such that for all fixed classical $x$ the nonclassical relation $R^{\text{NCL}}(x, b)$ holds.

(VI) Standardization for nonclassical sets

$$ \forall^{\text{st}}_{s,w} A^{\text{NCL}} \exists^{\text{st}} B^{\text{NCL}} \forall^{\text{st}}_{s,w} x^{\text{NCL}} (x \in_{s,w} B \Leftrightarrow_{s,w} x \in_{s,w} A \land \varphi(x)) \qquad (5.7) $$

for every st-$\epsilon$-formula $\varphi$ with arbitrary (internal) parameters.

(VII) Transfer for nonclassical sets

$$ \forall^{\text{st}}_{s,w} y_1^{\text{NCL}}, \dots, y_n^{\text{NCL}} \forall^{\text{st}} x^{\text{NCL}} [\varphi(x, y_1, \dots, y_n)] \Rightarrow_{s,w} \forall_{s,w} x^{\text{NCL}} \varphi(x, y_1, \dots, y_n) \qquad (5.8) $$

for all internal $\varphi(x, y_1, \dots, y_n)$.

**Boundedness for nonclassical sets**

$$ \forall_{s,w} x^{\text{NCL}} \exists^{\text{st}} y^{\text{NCL}} (x \in_{s,w} y) \qquad (5.9) $$

and since (5.9) contradicts idealization the following (bounded) form is taken instead:

(VIII) Bounded Idealization for nonclassical sets

For every $\epsilon$-formula $R$:

$$ \forall^{\text{st}}_{s,w} Y^{\text{NCL}} [\forall^{\text{stfin}}_{s,w} F^{\text{NCL}} \exists^{\text{y}} y^{\text{NCL}} \in_{s,w} Y (\forall_{s,w} x^{\text{NCL}} (x \in F) R(x, y) \Leftrightarrow_{s,w} \exists b^{\text{NCL}} (b \in Y) \forall^{\text{st}}_{s,w} x R(x, b))] \qquad (5.10) $$

(IX) Internal Hyper Infinite Induction

$$ \forall S (S \subset_s \mathbb{N}^\#) \{ \forall \beta (\beta \in \mathbb{N}^\#) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow_s \alpha^+ \in_s S) \right] \Rightarrow_s S =_s \mathbb{N}^\# \}. \qquad (5.11) $$

**The main restricted rules of conclusion.**

If $\text{IST}^\#$ $\vdash$ A then $\neg A \not\vdash B$, where $B \in \mathcal{L}^\#$.

Thus if statement A holds in $\text{IST}^\#$ we cannot obtain from $\neg A$ any formula B
whatsoever.

**Abbreviation 5.2** Let $X_n(A), \bar{X}_n(A) = n$ be a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n$. We shall abbreviated: $[X_n(A), (\bar{X}_n(A) = n)$ is a s-finite non-empty subset of a linearly s-ordered set $A$ such that there is a first element and a last element of $X_n(A)] \Leftrightarrow \hat{X}_n(A)$.

(X) **Axiom of existence non well-ordered s-finite subset of $\mathbb{N}$**.

$$ \exists m(m \in_s \mathbb{N}) \exists X_m(\mathbb{N}) [\neg \hat{X}_m(\mathbb{N})]. \qquad (5.12) $$

# §6. Hypernaturals $\mathbb{N}^#$. Axiom of hyperinfinity

**Definition 6.1.** (i) A non-empty transitive non regular set $u$ is a well formed non regular set iff:

(i) there is unique countable sequence $\{u_n\}_{n=1}^\infty$ such that

$$ \dots \in u_{n+1} \in u_n \dots \in u_4 \in u_3 \in u_2 \in u_1 \in u, \qquad (6.1) $$

(ii) for any $n \in \mathbb{N}$ and any $u_{n+1} \in u_n$ :

$$ u_n = u_{n+1}^+, \qquad (6.2) $$

where $a^+ = a \cup \{a\}$.

(ii) we define a function $a^{+[k]}$ inductively by $a^{+[k]} = (a^{+[k]})^+$.

**Definition 6.2.** Let $u$ and $w$ are well formed non regular sets. We write $w < u$ iff for any $n \in \mathbb{N}$

$$ w \in u_n. \qquad (6.3) $$

**Definition 6.3.** We say that an well formed non regular set $u$ is infinite (or hyperfinite) hypernatural number iff:

(I) For any member $w \in u$ one and only one of the following conditions are satisfied:

(i) $w \in \mathbb{N}$ or

(ii) $w = u_n$ for some $n \in \mathbb{N}$ or

(iii) $w < u$.

(II) Let $\prec u$ be a set $\prec u = \{z|z < u\}$, then by relation $(\cdot < \cdot)$ a set $\prec u$ is densely ordered with no first element.

(III) $\mathbb{N} \subset u$.

**Definition 6.4.** Assume $u \in \mathbb{N}^#$, then $u$ is infinite (hypernatural) number if $u \in \mathbb{N}^#\mathbb{N}$.

## Axiom of hyperinfinity

There exists a set $\mathbb{N}^#$ such that:

(i) $\mathbb{N} \subset \mathbb{N}^#,$

(ii) if $u \in \mathbb{N}^#\mathbb{N}$ then there exists infinite (hypernatural) number $v$ such that $v < u$,

(iii) if $u \in \mathbb{N}^#\mathbb{N}$ then there exists infinite (hypernatural) number $w$ such that for any $n \in \mathbb{N}: u^{+[n]} < w$,

(iv) set $\mathbb{N}^#\mathbb{N}$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

# §7. Axioms of the nonstandard arithmetic $A^#$.
Axioms of the nonstandard arithmetic $A^\#A$ are:

**Axiom of hyperinfinity**

There exists a set $N^\#$ such that:

(i) $N \subset N^\#$

(ii) if $u$ is infinite (hypernatural) number then there exists infinite (hypernatural) number $v$ such that $v < u$

(iii) if $u$ is infinite hypernatural number then there exists infinite (hypernatural) number $w$ such that $u < w$

(iv) set $N^\#N$ is patially ordered by relation $(\cdot < \cdot)$ with no first and no last element.

**Axioms of infinite $\omega$-induction**

(i)

$$ \forall S (S \subset N) \{ \left[ \bigwedge_{n \in \omega} (n \in S \Rightarrow_s n^+ \in S) \right] \Rightarrow_s S = N \}. \quad (7.1) $$

(ii) Let $F(x)$ be a wff of the set theory $NC_{\infty}^{\#\#}$, then

$$ \left[ \bigwedge_{n \in \omega} (F(n) \Rightarrow_s F(n^+)) \right] \Rightarrow_s \forall n (n \in \omega) F(n). \quad (7.2) $$

**Definition 7.1.** (i) Let $\beta$ be a hypernatural such that $\beta \in N^\#N$. Let $[0, \beta] \subset N^\#$ be a set such that $\forall x[x \in [0, \beta] \Leftrightarrow 0 \le x \le \beta]$ and let $[0, \beta)$ be a set $[0, \beta) = [0, \beta] \setminus \{\beta\}$.

(ii) Let $\beta \in N^\#N$ and let $\beta_\infty \subset N^\#$ be a set such that

$$ \forall x \{x \in \beta_\infty \Leftrightarrow \exists k (k \ge 0) [0 \le x \le \beta^{[k]}]\}. \quad (7.3) $$

**Definition 7.2.** Let $F(x)$ be a wff of $NC_{\infty}^{\#\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is restricted on a classical set $S$ such that $S \subsetneq_s N^\#$ iff the following condition is satisfied

$$ \forall \alpha[\alpha \in N^\#S \Rightarrow_s \neg F(\alpha)]. \quad (7.4) $$

**Definition 7.3.** Let $F(x)$ be a wff of $NC_{\infty}^{\#\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is *strictly* restricted on a set $S$ such that $S \subsetneq_s N^\#$ iff there is no proper subset $S' \subset S$ such that a wff $F(x)$ is restricted on a set $S'$.

**Example 7.1.** (i) Let $\mathbf{fin}(\alpha), \alpha \in N^\#$ be a wff formula such that $\mathbf{fin}(\alpha) \Leftrightarrow_s \alpha \in N$.

Obviously wff $\mathbf{fin}(\alpha)$ is strictly restricted on a set $N$ since $\forall \alpha[\alpha \in N^\#N \Rightarrow_s \neg \mathbf{fin}(\alpha)]$.

Let $\mathbf{hfin}(\alpha), \alpha \in N^\#$ be a wff formula such that $\mathbf{hfin}(\alpha) \Leftrightarrow_s \alpha \in N^\#N$ since

$$ \forall \alpha[\alpha \in N \Rightarrow_s \neg \mathbf{hfin}(\alpha)]. $$

**Definition 7.4.** Let $F(x)$ be a wff of $NC_{\infty}^{\#\#}$ with unique free variable $x$. We will say that a wff $F(x)$ is unrestricted if wff $F(x)$ is not restricted on any set $S$ such that $S \subsetneq N^\#$.

**Axiom of hyperfinite induction 1**

$$ \forall S (S \subseteq_s [0, \beta]) \forall \beta (\beta \in_s N^\#) \searrow \\ \left\{ \forall \alpha (\alpha \in_s [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in_s S \Rightarrow \alpha^+ \in_s S) \right] \Rightarrow_s S = [0, \beta] \right\}. \quad (7.5) $$

**Axiom of hyperfinite induction 1'**
$$ \forall S (S \subseteq_s [0, \beta_\infty]) \forall \beta (\beta \in \mathbb{N}^#) \searrow \\ \left\{ \forall \alpha (\alpha \in [0, \beta_\infty]) \left[ \bigwedge_{0 \le \alpha < \beta_\infty} (\alpha \in S \Rightarrow \alpha^+ \in S) \right] \Rightarrow S = [0, \beta_\infty] \right\}. \qquad (7.6) $$

**Axiom of hyper infinite induction 1**

$$ \forall S (S \subseteq_s \mathbb{N}^#) \left\{ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (\alpha \in S \Rightarrow \alpha^+ \in S) \right] \Rightarrow_S S =_S \mathbb{N}^# \right\}. \qquad (7.7) $$

**Definition 7.5.** A set $S \subseteq_s \mathbb{N}^#$ is a hyper inductive if the following statement holds

$$ \bigwedge_{\alpha \in \mathbb{N}^#} (\alpha \in_S S \Rightarrow_S \alpha^+ \in_S S). \qquad (7.8) $$

Obviously a set $\mathbb{N}^#$ is a hyper inductive. Thus axiom of hyper infinite induction 1 asserts that a set $\mathbb{N}^#$ this is the smallest hyper inductive set.

**Axioms of hyperfinite induction 2**

Let $F(x)$ be a wff of the set theory $\mathbf{NC}^#_{\infty}$ strictly restricted on a set $[0, \beta]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta]) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_S F(\alpha^+)) \right] \right] \Rightarrow_S \forall \alpha (\alpha \in [0, \beta]) F(\alpha). \qquad (7.9) $$

Let $F(x)$ be a wff of the set theory $\mathbf{NC}^#_{\infty}$ strictly restricted on a set $[0, \beta_\infty]$ then

$$ \left[ \forall \beta (\beta \in [0, \beta_\infty]) \left[ \bigwedge_{0 \le \alpha < \beta_\infty} (F(\alpha) \Rightarrow_S F(\alpha^+)) \right] \right] \Rightarrow_S \forall \alpha (\alpha \in [0, \beta_\infty]) F(\alpha). \qquad (7.10) $$

**Axiom of hyper infinite induction 2**

Let $F(x)$ be an unrestricted wff of the set theory $\mathbf{NC}^#_{\infty}$ then

$$ \left[ \forall \beta (\beta \in \mathbb{N}^#) \left[ \bigwedge_{0 \le \alpha < \beta} (F(\alpha) \Rightarrow_S F(\alpha^+)) \right] \right] \Rightarrow_S \forall \beta (\beta \in \mathbb{N}^#) F(\beta). \qquad (7.11) $$

**The main restricted rules of conclusion.**

If $A^# \vdash A$ then $\neg A \vdash B$, where $B \in \mathcal{L}^#$.

Thus if statement A holds in $A^#$ we cannot obtain from $\neg A$ any formula B whatsoever.

## §8. The Generalized Recursion Theorem.

**Theorem 1.** Let $S$ be a set, $c \in S$ and $G: S \to S$ is any function with $\text{dom}(G) = S$ and $\text{range}(G) \subseteq S$. Let $W[G] \in \mathbb{N}^# \times S$ be a binary relation such that:

(a) $(1, c) \in W[G]$ and

(b) if $(x, y) \in W[G]$ then $(\mathbf{Sc}(x), G(y)) \in W[G]$.

Then there exists a function $\mathcal{F}: \mathbb{N}^# \to S$ such that:

(i) $\text{dom}(\mathcal{F}) = \mathbb{N}^# \text{ and range}(\mathcal{F}) \subseteq S$;

(ii) $\mathcal{F}(1) = c$;

(iii) for all $x \in \mathbb{N}^#$, $\mathcal{F}(\mathbf{Sc}(x)) = G(\mathcal{F}(x))$.

1. The desired function $\mathcal{F}$ is a certain hyper inductive relation $W \subseteq \mathbb{N}^# \times S$. It is to have the properties:

(ii') $(1, c) \in W$;

(iii') if $(x, y) \in W$ then $(\mathbf{Sc}(x), G(y)) \in W$.

**Remark 1.** The latter is just another way of expressing (iii), that if
$$ \mathcal{F}(x) = y \qquad (1) $$

then

$$ \mathcal{F}(\mathbf{Sc}(x)) = G(y). \qquad (2) $$

**Remark 2.** Note that any relation $\mathbf{W}$ mentioned above is hyper inductive relation since the hyper inductivity conditions (ii')-(iii') are satisfied.

However there are many hyper inductive relations which satisfy the conditions (ii')-(iii'); on such is $\mathbb{N}^\# \times \mathcal{S}$. What distinguishes the desired function from all these other relations is that we want $(a, b)$ to be on it only as required by (ii') and (iii'). In other words, it is to be the smallest relation satisfying (ii')-(iii'). This can be expressed precisely as follows:

(1) Let $\mathbf{M}$ be a set of the relations $\mathbf{W}$ satisfying the conditions (ii') and (iii'); then we define

$$ \mathcal{F} = \bigcap_{\mathbf{W} \in \mathbf{M}} \mathbf{W}. $$

Hence

(2) whenever $\mathbf{W} \in \mathbf{M}$ then $\mathcal{F} \subseteq \mathbf{W}$.

We shall now show that we can derived from (1) that $\mathcal{F}$ is also one relation in $\mathbf{M}$.

(3) $(1, c) \in \mathcal{F}$.

This follows immediately from the definition of $\bigcap_{\mathbf{W} \in \mathbf{M}}$ and the fact that $(1, c) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$.

(4) If $(x, y) \in \mathcal{F}$ then $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$.

For if $(x, y) \in \mathcal{F}$ then $(x, y) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$; hence by (iii')

$(\mathbf{Sc}(x), G(y)) \in \mathbf{W}$ for all $\mathbf{W} \in \mathbf{M}$ so that $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$ by (1).

We must now verify that $\mathcal{F}$ is actually a function, i.e., we wish to show that for any $x, z_1, z_2 \in \mathbb{N}^\#$, if $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$, then $z_1 = z_2$.

We shall prove this by **hyper infinite induction** on $x$. Let

(5) $A = \{x | x \in \mathbb{N}^\# \text{ and for all } z_1, z_2 \in \mathbb{N}^\#, \text{ if } (x, z_1) \in \mathcal{F} \text{ and } (x, z_2) \in \mathcal{F} \text{ then } z_1 = z_2\}$.

We shall show $A = \mathbb{N}^\#$ by applying hyper infinite induction. First we have

(6) $1 \in A$.

To prove (6), it suffices to show that for any $z$, if $(1, z) \in \mathcal{F}$ then $z = c$.

We prove this by contradiction; in other words, suppose to be the contrary that there is some $z$ with $(1, z) \in \mathcal{F}$ but $z \neq c$. Consider the relation $W = \mathcal{F} \setminus \{(1, z)\}$. Since $(1, c) \in \mathcal{F}$ and $(1, c) \neq (1, z)$, it follows that $(1, c) \in \mathbf{W}$. Moreover, whenever $(u, y) \in \mathbf{W}$ then $(u, y) \in \mathcal{F}$ and hence $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$ but $\mathbf{Sc}(u) \neq 1$, so $(\mathbf{Sc}(u), G(y)) \neq (1, z)$, and hence $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$. Thus $\mathbf{W}$ satisfies both conditions (ii') and (iii'); in other words, $\mathbf{W} \in \mathbf{M}$. But then it follows from (2) that $\mathcal{F} \subseteq \mathbf{W}$ however this is clearly false since $(1, z) \in \mathcal{F}$ and $(1, z) \notin \mathbf{W}$. Thus our hypothesis has led us to a contradiction, and hence (6) is proved. Next we show that

(7) for any $x \in \mathbb{N}^\#$ if $x \in A$ then $\mathbf{Sc}(x) \in A$.

Suppose that $x \in A$, so that whenever $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$ then $z_1 = z_2$. We must show that whenever $(\mathbf{Sc}(x), w_1) \in \mathcal{F}$ and $(\mathbf{Sc}(x), w_2) \in \mathcal{F}$ then $w_1 = w_2$. To prove this, it suffices to show that

(8) whenever $(\mathbf{Sc}(x), w) \in \mathcal{F}$ then there exists a $z$ with $w = G(z)$ and $(x, z) \in \mathcal{F}$.
For if (8) is true, we would have for the given $w_1, w_2$ some $z_1 = z_2$ with
$w_1 = G(z_1)$, $w_2 = G(z_2)$, $(x, z_1) \in \mathcal{F}$ and $(x, z_2) \in \mathcal{F}$. Then, since $x \in A, z_1 = z_2$
and hence $G(z_1) = G(z_2)$, that is, $w_1 = w_2$.

Now to prove (8) suppose, to the contrary, that it is not true; in other words,
suppose that we have some $w$ with $(\mathbf{Sc}(x), w) \in \mathcal{F}$ but such that for all
$z$ which $(x,z) \in \mathcal{F}$ we have $w \neq G(z)$. Consider the relation $\mathbf{W} = \mathcal{F}\setminus\{(\mathbf{Sc}(x), w)\}$.

We shall show that $\mathbf{W} \in \mathbf{M}$. First of all $(1, c) \in \mathcal{F}$ and $(1, c) \neq (\mathbf{Sc}(x), w)$; hence
$(1, c) \in \mathbf{W}$. Suppose that $(u, y) \in \mathbf{W}$; then $(u, y) \in \mathcal{F}$ and $(\mathbf{Sc}(u), G(y)) \in \mathcal{F}$.

Clearly if $u \neq x$ then $(\mathbf{Sc}(u), G(y)) \neq (\mathbf{Sc}(x), w)$, so that in this case $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$.

On the other hand, if $u = x$ and $(\mathbf{Sc}(u), G(y)) = (\mathbf{Sc}(x), w)$, then $w = G(y)$, where $(x,y) \in \mathcal{F}$, contrary to the choice of $w$ henee $(\mathbf{Sc}(u), G(y)) \neq (\mathbf{Sc}(x), w)$, so again $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$. Thus whenever $(u,y) \in \mathbf{W}$, also $(\mathbf{Sc}(u), G(y)) \in \mathbf{W}$. Now that we have shown $\mathbf{W} \in \mathbf{M}$ we see by (2) that $\mathcal{F} \subseteq \mathbf{W}$ but this is false since $(\mathbf{Sc}(x), w) \in \mathcal{F}$ and $(\mathbf{Sc}(x), w) \notin \mathbf{W}$. Thus our hypothesis that (8) is incorrect has led to a contradiction, and now (8) is proved. Since (7) follows from (8), we have by *hyper infinite induction* from (6) that $A = \mathbb{N}^#$. Hence

(9) $\mathcal{F}$ is a function.

We have still to prove that $\mathcal{F}$ satisfies, condition (i); we must show that
for each $x \in \mathbb{N}^#$ there is a $y$ with $(x,y) \in \mathcal{F}$. Since $\mathcal{F} \subseteq \mathbb{N}^# \times S$, it will
then follow that $\mathrm{dom}(\mathcal{F}) = \mathbb{N}^# \text{ and } \mathrm{range}(\mathcal{F}) \subseteq S$. Let $B = \mathrm{dom}(\mathcal{F})$, that is,
(10) $B = \{x|x \in \mathbb{N}^# \text{ and for some } y, (x,y) \in \mathcal{F}\}$.

We prove now by **hyper infinite induction** that $B = \mathbb{N}^#$. First, $1 \in B$, sínce $(1, c) \in \mathcal{F}$
by (3). Next, if $x \in B$, pick some $y$ with $(x,y) \in \mathcal{F}$; then by (4), $(\mathbf{Sc}(x), G(y)) \in \mathcal{F}$,
and henee $\mathbf{Sc}(x) \in B$.

Thus concludes the first part of the proof, that there is at least one function $\mathcal{F}$
satisfying conditions (i)-(iii).

**Part 2.** We prove that there cannot be more than one such function.

Suppose that $\mathcal{F}_1$ and $\mathcal{F}_2$ both satisfy the conditions (i)-(iii) we wish to show
$\mathcal{F}_1 = \mathcal{F}_2$, i.e., that for all $x \in \mathbb{N}^#$, $\mathcal{F}_1(x) = \mathcal{F}_2(x)$. Thus
is proved by **hyper infinite induction** on $X$. By (ii), $\mathcal{F}_1(1) = c$ and $\mathcal{F}_2(1) = c$, so
$\mathcal{F}_1(1) = \mathcal{F}_2(1)$. Suppose that $\mathcal{F}_1(x) = \mathcal{F}_2(x)$; then $\mathcal{F}_1(\mathbf{Sc}(x)) = G(\mathcal{F}_1(x))$
and $\mathcal{F}_2(\mathbf{Sc}(x)) = G(\mathcal{F}_2(x))$, so $\mathcal{F}_1(\mathbf{Sc}(x)) = \mathcal{F}_2(\mathbf{Sc}(x))$.

**Theorem 2.** Let $S$ be a set, $c \in S$ and $G: S \times \mathbb{N}^# \to S$ is a binary function with
$\mathbf{dom}(G) = S \times \mathbb{N}^# \text{ and } \mathbf{range}(G) \subseteq S$.

Then there exists a function $\mathcal{F}: \mathbb{N}^# \to S$ such that:

(i) $\mathbf{dom}(\mathcal{F}) = \mathbb{N}^# \text{ and } \mathbf{range}(\mathcal{F}) \subseteq S$;

(ii) $\mathcal{F}(1) = c$;

(iii) for all $x \in \mathbb{N}^#$, $\mathcal{F}(\mathbf{Sc}(x)) = G(\mathcal{F}(x), x)$.

We omit the proof of the Theorem 3.4.2 since it can be given by simple modification
of the proof to Theorem 3.4.1.

§9. General associative and commutative laws.

**Definition C.1.** Suppose that $S$ is a set on which a binary operation $+$ is defined and
under which $S$ is closed. Let $\{x_k\}_{k \in \mathbb{N}^#}$ be any hyper infinite sequence of terms of $S$. For
every $n \in \mathbb{N}^#$ we denote by $\mathrm{Ext}-\sum_{k=1}^{n} x_k$ the element of $S$ uniquely determined by the
following conditions:

$$
(i) \operatorname{Ext}-\sum_{k=1}^{1} x_{k}=x_{1} ;
$$

Remark 9.1. This definition is justified on the following grounds. The sequence
{$x_k$}$_{k \in \mathbb{N}^\#}$ is a given external function $H$ with domain $\mathbb{N}^\#$, $H(x_k) = x_k$ for every $k$. We seek
a function $F$ with domain $\mathbb{N}^\#$ whose value $F(n)$ is to be $\operatorname{Ext}-\sum_{k=1}^n x_k$. Then the conditions

(i), (ii) above correspond to the following conditions on F :

$$
(i') F(1) = H(1); (ii') F(n+1) = F(n) + H(n+1), \text{ for all } n \in \mathbb{N}^{\#}.
$$

Let (1) $c = H(1)$; (2) $G(n, z) = z + H(n + 1)$.

Thus the conditions (i') and (ii') are equivalent to

$$
(i'') F(1) = c;
$$

$$
(ii'') F(n + 1) = G(n, F(n)) \text{ for all } n \in \mathbb{N}^{\#}.
$$

Given the function *H*, the element *c* of *S* and the function *G* are well-defined by (1)-(2).

Then by Theorem B.1 we see that there is a unique function *F* satisfying (1)-(2) with
**dom**(*F*) = **N**\# and **range**(*F*) ⊆ *S*. Thus (*i*')-(*ii*') is just another form of recursive
definition.

(Hence it should be expected that various properties of $\operatorname{Ext}-\sum_{k=1}^{n} x_k$ will have to be
verified
by hyper infinite induction on $n \in \mathbb{N}^{\#}$.)

**Definition 9.2.** Suppose that *S* is a set on which a binary operation $\times$ is defined and under which *S* is dosed. Let $\{x_k\}_{k \in \mathbb{N}^\#}$ be an hyper infinite sequence of terms of *S*. For every $n \in \mathbb{N}^\#$ we denote by $\operatorname{Ext}-\prod_{k=1}^n x_k$ the element of *S* uniquely determined by the following conditions:

$$
(i) \operatorname{Ext}-\prod_{k=1}^{n} x_k = x_1; \quad (ii) \operatorname{Ext}-\prod_{k=1}^{n+1} x_k = \left(\operatorname{Ext}-\prod_{k=1}^{n} x_k\right) \times x_{n+1} \quad \text{for all } n \in \mathbb{N}^{\#}.
$$

**Theorem 1. (1)** Suppose that *S* is a set closed under a binary operation $+$ and that $+$ is associative on *S*, i.e., for all *x*,*y*,*z* ∈ *S*, *x* + (*y* + *z*) = (*x* + *y*) + *z*. Let {$x_k$}$_{k \in \mathbb{N}^\#}$ be any hyper infinite sequence of terms in *S*. Then for any *n*,*m* ∈ $\mathbb{N}^\#$. we have

$$
\operatorname{Ext}-\sum_{k=1}^{n+m} x_k = \left( \operatorname{Ext}-\sum_{k=1}^{n} x_k \right) + \left( \operatorname{Ext}-\sum_{k=1}^{m} x_{n+k} \right). \qquad (9.1)
$$

(2) Suppose that $S$ is a set closed under a binary operation $\times$ and that $\times$ is associative on $S$, i.e., for all $x, y, z \in S$, $x \times (y \times z) = (x \times y) \times z$. Let $\{x_k\}_{k \in \mathbb{N}^\#}$ be any hyper infinite sequence of terms in $S$. Then for any $n, m \in \mathbb{N}^\#$. we have

$$
\operatorname{Ext}-\prod_{k=1}^{n+m} x_k = \left( \operatorname{Ext}-\prod_{k=1}^{n} x_k \right) \times \operatorname{Ext}-\prod_{k=1}^{m} x_{n+k}. \quad (9.2)
$$

**Proof.** We prove (3.5.1); the proof of (2) is completely similar. Let *n* be fixed; we proceed by hyper infinite induction on *m*. For *m* = 1 from Eq.(3.8.1) we get

$$
\operatorname{Ext}-\sum_{k=1}^{n+1} x_k = \left( \operatorname{Ext}-\sum_{k=1}^{n} x_k \right) + \left( \operatorname{Ext}-\sum_{k=1}^{1} x_{n+k} \right). \quad (9.3)
$$

By Definition 3.8.1(i) we obtain
$$ \mathrm{Ext}-\sum_{k=1}^{1} x_{n+k} = x_{n+1}. \qquad (9.4) $$

Suppose Eq.(3.8.1) is true for $m \in \mathbb{N}^#$. We show that is true for $m+1$, i.e., that

$$ \mathrm{Ext}-\sum_{k=1}^{n+(m+1)} x_k = \left( \mathrm{Ext}-\sum_{k=1}^{n} x_k \right) + \left( \mathrm{Ext}-\sum_{k=1}^{m+1} x_{n+k} \right). \qquad (9.4') $$

By associativity + on $\mathbb{N}^#$ we get

$$ \mathrm{Ext}-\sum_{k=1}^{n+(m+1)} x_k = \mathrm{Ext}-\sum_{k=1}^{(n+m)+1} x_k. \qquad (9.6) $$

From Eq.(3.8.6) by Definition 3.8.1(ii) we obtain

$$ \mathrm{Ext}-\sum_{k=1}^{(n+m)+1} x_k = \mathrm{Ext}-\sum_{k=1}^{n+m} x_k + x_{(n+m)+1} = \mathrm{Ext}-\sum_{k=1}^{n+m} x_k + x_{n+(m+1)}. \qquad (9.7) $$

From Eq.(3.8.7) by induction hypothesis we obtain

$$ \mathrm{Ext}-\sum_{k=1}^{n+m} x_k + x_{n+(m+1)} = \left( \mathrm{Ext}-\sum_{k=1}^{n} x_k + \mathrm{Ext}-\sum_{k=n}^{m} x_k \right) + x_{n+(m+1)}. \qquad (9.8) $$

From Eq.(3.8.8) by associativity + on $S$ we get

$$ \left( \mathrm{Ext}-\sum_{k=1}^{n} x_k + \mathrm{Ext}-\sum_{k=n}^{m} x_k \right) + x_{n+(m+1)} = \mathrm{Ext}-\sum_{k=1}^{n} x_k + \left( \mathrm{Ext}-\sum_{k=n}^{m} x_k + x_{n+(m+1)} \right). \qquad (9.9) $$

From Eq.(3.8.9) by Definition 3.8.1(ii) we obtain

$$ \mathrm{Ext}-\sum_{k=1}^{n} x_k + \left( \mathrm{Ext}-\sum_{k=n}^{m} x_k + x_{n+(m+1)} \right) = \mathrm{Ext}-\sum_{k=1}^{n} x_k + \mathrm{Ext}-\sum_{k=n}^{m+1} x_k. \qquad (9.10) $$

This equality completes the inductive step and hence the proof of the theorem.

**Definition 9.3.** Let $\langle x_1, \dots, x_n \rangle, n \in \mathbb{N}^# \mathbb{N}$ be an hyperfinite sequence of elements of $\mathbb{R}_c^#$.

Then $\mathrm{Ext}-\sum_{k=m}^{n} x_k$ and $\mathrm{Ext}-\prod_{k=m}^{n} x_k$ are defined for any $n, m \in \mathbb{N}^#$ by the recursions

(i) $\mathrm{Ext}-\sum_{k=m}^{n} x_k = 0$ and $\mathrm{Ext}-\prod_{k=m}^{n} x_k = 1$ if $n < m$;

(ii) $\mathrm{Ext}-\sum_{k=m}^{n} x_k = \left( \mathrm{Ext}-\sum_{k=m}^{n-1} x_k \right) + x_n$ and

(iii) $\mathrm{Ext}-\prod_{k=m}^{n} x_k = x_n \times \left( \mathrm{Ext}-\prod_{k=m}^{n-1} x_k \right)$ if $m < n$.

The condition (ii) of the above definition is justified by recursive definition, see Appendix B.

**Definition 9.4.** Let $\langle x_1, \dots, x_j, \dots \rangle, j \in \mathbb{N}$ be a countable sequence of elements of $\mathbb{R}_c^#$.

Then $\omega$-sum $\mathrm{Ext}-\sum_{j=m}^{\omega} x_k$ and $\omega$-product $\mathrm{Ext}-\prod_{j=m}^{\omega} x_k$ are defined for any $m \in \mathbb{N}$ by

(iv) $\mathrm{Ext}-\sum_{j=m}^{\omega} x_j \triangleq \mathrm{Ext}-\sum_{j=m}^{n} y_j$, where $\langle y_1, \dots, y_j, \dots, y_n \rangle, n \in \mathbb{N}^# \mathbb{N}$ is a hyperfinite sequence such that $x_j = y_j$ for all $j \in \mathbb{N}$ and $y_j = 0$ for all $j \in \mathbb{N}^# \mathbb{N}$;

(v) $\mathrm{Ext}-\prod_{j=m}^{\omega} x_j \triangleq \mathrm{Ext}-\prod_{j=m}^{\omega} y_j$, where $\langle y_1, \dots, y_j, \dots, y_n \rangle, n \in \mathbb{N}^# \mathbb{N}$ is a hyperfinite sequence such that $x_j = y_j$ for all $j \in \mathbb{N}$ and $y_j = 1$ for all $j \in \mathbb{N}^# \mathbb{N}$.
**Theorem 9.2.** Let $\langle x_1, \dots, x_n \rangle, n \in \mathbb{N}^\# \setminus \mathbb{N}$ be an hyperfinite sequence of elements of $\mathbb{R}_c^\#$. Then we have

$$ \mathrm{Ext}-\sum_{k=m}^{n} x_k = \mathrm{Ext}-\sum_{k=m}^{n-m+q} x_{k+m-q} \qquad (9.11) $$

and

$$ z \times \left( \mathrm{Ext}-\sum_{k=m}^{n} x_k \right) = \mathrm{Ext}-\sum_{k=m}^{n} z \times x_k, \qquad (9.12) $$

$z \in \mathbb{R}_c^\#.$

**Proof.** Let $\langle x_1, \dots, x_n \rangle, n \in \mathbb{N}^\# \setminus \mathbb{N}$ be an hyperfinite sequence of elements of $\mathbb{R}_c^\#$. Consider now any hyperfinite nonnegative integers

$n_1, n_2, \dots, n_i, \dots, n_t, n_i \in \mathbb{N}^\# \setminus \mathbb{N}, 1 \le i \le t,$

and set

$$ n = n_1 + n_2 + \dots + n_t. \qquad (9.13) $$

Given $x_1, \dots, x_n$, we can group these as:

$$ x_1, \dots, x_{n_1}; \ x_{n_1+1}, \dots, x_{n_1+n_2}; \ x_{n_1+n_2+1}, \dots, x_{n_1+n_2+n_3}; \dots x_{n_1+n_2+\dots+n_{i+1}}, \dots, x_{n_1+n_2+\dots+n_{i+1}}; \dots \qquad (9.14) $$

Here, if $n_i = 0$, the corresponding subsequence is regarded as being empty.

**Theorem 9.3.** Let $\langle x_1, \dots, x_k, \dots \rangle$ be an hyper infinite sequence of elements of $\mathbb{R}_c^\#$. Let $\langle n_1, \dots, n_t \rangle$ be a sequence of nonnegative integers. For each $i = 1, \dots, t \in \mathbb{N}^\#$,

let $m_i = \sum_{j=1}^{i-1} n_j$ and let $n = m_t + n_t$. Then

$$ \mathrm{Ext}-\sum_{k=1}^{n} x_k = \sum_{i=1}^{t} \left( \mathrm{Ext}-\sum_{k=1}^{n_i} x_{m_i+k} \right) \qquad (9.15) $$

and

$$ \mathrm{Ext}-\prod_{k=1}^{n} x_k = \prod_{i=1}^{t} \left( \mathrm{Ext}-\prod_{k=1}^{n_i} x_{m_i+k} \right). \qquad (9.16) $$

**Proof.** By hyper infinite induction.

**Definition 9.5.** A function $F$ is said to be a permutation of a set $S$ if it is one-to-one and $\mathrm{dom}(F) = \mathrm{range}(F) = S$.

**Definition 9.6.** Let $[1, n]$ a set $\{k|k \in \mathbb{N}^\# \wedge (1 \le k \le n)\}$

**Theorem 9.4.** Let $\langle x_1, \dots, x_n \rangle, n \in \mathbb{N}^\# \setminus \mathbb{N}$ be an hyperfinite external sequence of elements of $\mathbb{R}_c^\#$. Then for any $n \in \mathbb{N}^\#$ and any permutation $F$ of $[1, n]$ following holds

$$ \mathrm{Ext}-\sum_{k=1}^{n} x_k = \mathrm{Ext}-\sum_{k=1}^{n} x_{F(k)}. \qquad (9.17) $$

The same holds if we replace $\mathrm{Ext}-\sum$ by $\mathrm{Ext}-\prod$.

**Proof.** The proof is by hyper infinite induction on $n \in \mathbb{N}^\#$. For $n = 1$ it is trivial.

Suppose that it is true for $n$. Let $G$ be a permutation of $[1, n+1]$. Then $G(m) = n+1$ for a unique $m$, such that $1 \le m \le n+1$. Then by Eq.(3.5.15)

$$ \mathrm{Ext}-\sum_{k=1}^{n+1} x_{G(k)} = \mathrm{Ext}-\sum_{k=1}^{m-1} x_{G(k)} + x_{n+1} + \mathrm{Ext}-\sum_{k=m+1}^{n+1} x_{G(k)} \qquad (9.18) $$

and by Eq.(3.8.18)
$$ Ext-\sum_{k=1}^{m-1} x_{G(k)} + x_{n+1} + Ext-\sum_{k=m+1}^{n+1} x_{G(k)} = Ext-\sum_{k=1}^{m-1} x_{G(k)} + Ext-\sum_{k=m}^{n} x_{G(k+1)} + x_{n+1}. \quad (9.19) $$

Thus by Eq.(3.8.11) we obtain

$$ Ext-\sum_{k=1}^{n+1} x_{G(k)} = Ext-\sum_{k=1}^{m-1} x_{G(k)} + Ext-\sum_{k=m}^{n} x_{G(k+1)} + x_{n+1}. \quad (9.20) $$

To reduce this to the inductive hypothesis, we wish to rewrite the external sum of the first

two terms as $Ext-\sum_{k=1}^{n} x_{F(k)}$ for suitable $F$. Define $F$ by

$$ F(k) = \begin{cases} G(k) & \text{if } 1 \le k < m \\ G(k+1) & \text{if } m \le k \le n \end{cases} \quad (9.21) $$

Since all values of $G(k)$ for $k \ne m$, we have for all $k \le n$

$$ 1 \le F(k) \le n \quad (9.22) $$

Now we claim that

$$ F \text{ is a permutation of } [1, n]. \quad (9.23) $$

By (3.8.21) and (3.8.22) we need only check that $F$ is one-to one. Suppose that $F(k_1) = F(k_2)$.

If both $k_1, k_2$ are $< m$ or both are $\ge m$, it follows from (3.8.21) and the fact that $G$ is a permutation that $k_1 = k_2$. If, say, $k_1 < m \le k_2$, we have $G(k_1) = G(k_2 + 1)$, hence $k_1 = k_2 + 1$, which contradicts our assumption. Thus neither this case nor, by symmetry, the case $k_2 < m \le k_1$ can occur. We have from (3.8.20) and (3.8.21) that

$$ Ext-\sum_{k=1}^{m+1} x_{G(k)} = Ext-\sum_{k=1}^{m-1} x_{F(k)} + Ext-\sum_{k=m}^{n} x_{F(k)} + x_{n+1} = Ext-\sum_{k=1}^{n} x_{F(k)} + x_{n+1} \quad (9.24) $$

by (3.8.23) and inductive hypothesis

$$ Ext-\sum_{k=1}^{n} x_{F(k)} + x_{n+1} = Ext-\sum_{k=1}^{n} x_{k} + x_{n+1} = Ext-\sum_{k=1}^{n+1} x_{k} \quad (9.25) $$

This equality completes the inductive step and hence the proof of the theorem.

# Chapter I. $\mathbb{R}^\#$-Valued abstract measures

## 1. $\sigma^\#$-algebras

**Definition 1.1** ($\sigma^\#$-algebra). Let $X$ be any set. We denote by $2^X = P(X) = \{A : A \subset X\}$ the set of all subsets of $X$. A family $\mathcal{F} \subset 2^X$ is called a $\sigma^\#$-algebra (on $X$) if:

(i) $\emptyset \in \mathcal{F}$;

(ii) $\mathcal{F}$ is closed under complements, i.e. $A \in \mathcal{F}$ implies $X \setminus A \in \mathcal{F}$;

(iii) $\mathcal{F}$ is closed under hypercountable unions, i.e. if $(A_n)_{n \in \mathbb{N}^\#}$ is a hyper infinite sequence in $\mathcal{F}$ then $\bigcup_{n \in \mathbb{N}^\#} A_n \in \mathcal{F}$.

**Proposition 1.1.** If $\mathcal{F}$ is a $\sigma^\#$-algebra on $X$ then:

1. $\mathcal{F}$ is closed under hypercountable intersections, i.e. if $(A_n)_{n \in \mathbb{N}^\#}$ is a hyper infinite
sequence in $\mathcal{F}$ then $\bigcap_{n \in \mathbb{N}^\#} A_n \in \mathcal{F}$.

2. $X \in \mathcal{F}$.

3. $\mathcal{F}$ is closed under hyperfinite unions and hyperfinite intersections.

4. $\mathcal{F}$ is closed under set differences.

5. $\mathcal{F}$ is closed under symmetric differences.

**Proposition 1.2.** Suppose $\mathcal{F} \subset 2^X$ is a family of subsets satisfying the following:

1. $\emptyset \in \mathcal{F}$;

2. $\mathcal{F}$ is closed under complements;

3. $\mathcal{F}$ is closed under hyperinfinite intersections.

Then $\mathcal{F}$ is a $\sigma^\#$-algebra.

**Proposition 1.3.** If $(\mathcal{F}_\alpha)_{\alpha \in I}$ is a collection of $\sigma^\#$-algebras on $X$, then $\bigcap_\alpha \mathcal{F}_\alpha$ is also a $\sigma^\#$-algebra on $X$.

**Proposition 1.4.** ($\sigma^\#$-algebra generated by subsets). Let $K$ be a collection of subsets of $X$. There exists a $\sigma^\#$-algebra, denoted $\sigma^\#(K)$ such that $K \subset \sigma^\#(K)$ and for every other $\sigma^\#$-algebra $\mathcal{F}$ such that $K \subset \mathcal{F}$ we have that $\sigma^\#(K) \subset \mathcal{F}$
We call $\sigma^\#(K)$ the $\sigma^\#$-algebra generated by $K$.

**Proof.** Define $\sigma^\#(K) \triangleq \bigcap \{\mathcal{F} \mid \mathcal{F} \text{ is a } \sigma^\#$-algebra on } X, K \subset \mathcal{F}\}$. This is a $\sigma^\#$-algebra with the required properties.

**Proposition 1.5.** If $K \subset \mathcal{L}$ then $\sigma^\#(K) \subset \sigma^\#(\mathcal{L})$. Also, if $K \subset \mathcal{F}$ and $\mathcal{F}$ is a $\sigma^\#$-algebra, then $\sigma^\#(K) \subset \mathcal{F}$.

**Definition 1.2.** (Borel $\sigma^\#$-algebra). Given a topological space $X$, the Borel $\sigma^\#$-algebra is the $\sigma^\#$-algebra generated by the open sets. It is denoted $B^\#(X)$.

Specifically in the case $X = *\mathbb{R}_c^{\#d}$, $d \in \mathbb{N}^\#$ we have that
$$B_d^\# \triangleq B^\#(*\mathbb{R}_c^{\#d}) = \sigma^\#(U|U \text{ is an } \#\text{-open set}).$$

A Borel-#-measurable set, i.e. a set in $B^\#(X)$, is called a #-Borel set.

**Measurable functions.** Let $f$ be a $*\mathbb{R}_c^\#$-valued function defined on a set $X$. We suppose that some $\sigma^\#$-algebra $\Omega \subseteq P(X)$ is fixed.

**Definition 1.3.** We say that $f$ is #-measurable, if $f^{-1}([a, b]) \in \Omega$ for any hyperreals $a, b \in *\mathbb{R}_c^\#$ such that $a < b$.

The following three propositions are obvious.

**Proposition 1.7.** Let $f: X \to *\mathbb{R}_c^\#$ be a function. Then the following conditions are equivalent:

(a) $f$ is #-measurable;

(b) $f^{-1}([0, b)) \in \Omega$ for any hyperreal $b \in *\mathbb{R}_c^\#$;

(c) $f^{-1}((b, \infty)) \in \Omega$ for any hyperreal $b \in *\mathbb{R}_c^\#$;

(d) $f^{-1}(B) \in \Omega$ for any $B \in B(R)$.

**Proposition 1.8** Let $f$ and $g$ be #-measurable functions, then

(a) $\alpha \times f + \beta \times g$ is #-measurable for any $\alpha, \beta \in *\mathbb{R}_c^\#$;

(b) functions $\max\{f, g\}$ and $f \times g$ are #-measurable.

In particular, functions $f^+ := \max\{f, 0\}$, $f^- := (-f)^+$, and $|f| := f^+ + f^-$ are
\#-measurable.

## §2. #-Measures and measure #-space

**Definition 2.1.** A pair $(X, \mathcal{F})$ where $\mathcal{F}$ is a $\sigma^\#$-algebra on $X$ is called a #-measurable space. Elements of $\mathcal{F}$ are called #-measurable sets.
Given a #-measurable space $(X, \mathcal{F})$, a function $\mu^# : \mathcal{F} \to [0, \infty^#]$ is called a #-measure on $(X, \mathcal{F})$ if

1. $\mu^#(\emptyset) = 0$;

2. (Hyper infinite additivity) For all hyper infinite sequences $(A_n)_{n \in \mathbb{N}^#} \subset \mathcal{F}$ of pairwise disjoint sets in $\mathcal{F}$, we have that $\mu^#\left(\bigcup_{n \in \mathbb{N}^#} A_n\right) = \mathrm{Ext} - \sum_{n \in \mathbb{N}^#} \mu^#(A_n)$.

$(X, \mathcal{F}, \mu^#)$ is called a #-measure space.

**Definition 2.2.** A measure space $(X, \mathcal{F}, \mu^#)$ is called: (a) hyperfinite if $\mu^#(X) < \infty^#$.

(b) It is called $\sigma^$-hyperfinite if $X = \bigcup_{n \in \mathbb{N}^#} A_n$ where $A_n \in \mathcal{F}$ and $\mu^#(A_n) < \infty^#$ for all $n \in \mathbb{N}^#$.

**Definition 2.3.** Let $\Sigma$ be a $\sigma^$-algebra of subsets of a set $X$, and let $E = (E, \|\cdot\|_\#)$ be a non-Archimedean Banach space. A function $\mu^# : \Sigma \to E \cup \{\infty\}$ is called a vector-valued #-measure (or $E$-valued measure) if

(a) $\mu^#(\emptyset) = 0$;

(b) $\mu^#\left(\bigcup_{n \in \mathbb{N}^#} A_n\right) = \mathrm{Ext} - \sum_{n \in \mathbb{N}^#} \mu^#(A_n)$ for any pairwise disjoint sequence $A_n, n \in \mathbb{N}^#$, $A_n \subseteq \Sigma$;

(c) for any $S \in \Sigma$, $\mu^#(S) = \infty$, there exists $B \in \Sigma$ such that $B \subseteq S$ and $0 < \|\mu^#(B)\|_\# < \infty$.

**Definition 2.4.** (a) A function $\mu^# : \mathcal{F} \to \ast\mathbb{C}_c^\# \cup \{\infty\}$ is called a complex #-measure if

1. $\mu^#(\emptyset) = 0$,

2. $\mu^#\left(\bigcup_{n \in \mathbb{N}^#} A_n\right) = \mathrm{Ext} - \sum_{n \in \mathbb{N}^#} \mu^#(A_n)$ for any sequence $A_n, n \in \mathbb{N}^#$ of pairwise disjoint sets from $\mathcal{F}$, and, for any $A \in \mathcal{F}$, $\mu^#(A) = \infty$, there exists $B \in \mathcal{F}$ such that $B \subseteq A$ and $0 < |\mu^#(B)|_\# < \infty$.

(b) A function $\mu^# : \mathcal{F} \to \ast\mathbb{R}_c^\# \cup \{\infty\}$ is called a signed #-measure if $\mu^#(\emptyset) = 0$

$\mu^#\left(\bigcup_{n \in \mathbb{N}^#} A_n\right) = \mathrm{Ext} - \sum_{n \in \mathbb{N}^#} \mu^#(A_n)$ for any sequence $A_n, n \in \mathbb{N}^#$ of pairwise disjoint sets from $\mathcal{F}$, and, for any $A \in \mathcal{F}$, $\mu^#(A) = \infty$, there exists $B \in \mathcal{F}$ such that $B \subseteq A$ and $0 < |\mu^#(B)| < \infty$.

**Definition 2.5.** If a certain property involving the points of #-measure space is true, except a subset having #-measure zero, then we say that this property is true #-almost everywhere (abbreviated as #-a.e.).

**Proposition 2.5.** Let $\mu^#$ be a #-measure on a $\sigma^$-algebra $\mathcal{F}, A_n \in \mathcal{F}$, and $A_n \to A$. Then $A \in \mathcal{F}$ and $\mu^#(A) = \mathrm{lim}_{n \to \infty} \mu^#(A_n)$. In particular, if $(B_n)_{n=1}^\infty$ is a decreasing hyper infinite sequence of elements of $\mathcal{F}$ such that $\bigcap_{n=1}^\infty B_n = \emptyset$, then $\mu^#(B_n) \to 0$.

**Definition 2.6.** If $\mathcal{F}$ is a $\sigma^$-algebra of subsets of $X$ and $\mu^#$ is a #-measure on $\mathcal{F}$, then the triple $(X, \mathcal{F}, \mu)$ is called a #-measure space. The sets belonging to $\mathcal{F}$ are called #-measurable sets because the #-measure is defined for them.

## §2.1. #-Convergence of functions and the generalized
Egoroff theorem.

**Definition 2.1.1.** Let $f_n, n \in \mathbb{N}^\#$ be a hyper infinite sequence of $^*\mathbb{R}_c^\#$-valued functions defined on $X$. We say that:

1. $f_n \to_{\#} f$ pointwise, if $f_n(x) \to_{\#} f(x)$ for all $x \in X$;

2. $f_n \to_{\#} f$ almost #-everywhere (#-a.e.), if $f_n(x) \to_{\#} f(x)$ for all $x \in X$ except a set of #-measure 0;

3. $f_n \to_{\#} f$ uniformly, if for any $\varepsilon > 0, \varepsilon \approx 0$ there is $n(\varepsilon)$ such that $\sup\{|f_n(x) - f(x)|: x \in X\} \le \varepsilon$ for all $n \ge n(\varepsilon)$.

**Theorem 2.1.1.** (generalized Egoroff's theorem) Suppose that $\mu^\#(X) < *^\infty$, $\{f_n\}_{n=1}^* \to_{\#} f$ are #-measurable functions on $X$ such that $f_n \to_{\#} f$ #-a.e. Then, for every $\varepsilon \approx 0, \varepsilon > 0$, there exists $E \subseteq X$ such that $\mu^\#(E) < \varepsilon$ and $f_n \to_{\#} f$ uniformly on $E^c = X\setminus E$.

**Proof:** Without loss of generality, we may assume that $f_n \to_{\#} f$ everywhere on $X$ and (by replacing $f_n$ with $f_n - f$) that $f \equiv 0$. For $k, n \in *\mathbb{N}$, let

$E_n(k) := \bigcup_{m=n}^{*\infty} \{x : |f_m(x)| \ge k - 1\}$. Then, for a fixed $k, E_n(k)$ decreases as $n$ increases, and $\bigcap_{n=1}^{*\infty} E_n(k) = \emptyset$. Since $\mu^\#(X) < *\infty$, we conclude that $\mu^\#(E_n(k)) \to_{\#} 0$ as $n \to *\infty$.

Given $\varepsilon \approx 0, \varepsilon > 0$ and $*\mathbb{N}$, choose $n_k$ such that $\mu^\#(E_n(k)) < \varepsilon \times 2^{-k}$, and set

$E = \bigcup_{n=1}^{*\infty} E_n(k)$. Then $\mu^\#(E) < \varepsilon$, and we have $|f_n(x)| < k^{-1}(\forall n > n_k, x \notin E)$.

Thus $f_n \to_{\#} 0$ uniformly on $X\setminus E$.

Generalized exhaustion argument.

Let $(X, \Sigma, \mu^\#)$ be a $\sigma^\#$-finite #-measure space. Given a hyper infinite sequence $(U_n)_n^\infty \subseteq \Sigma$, a set $A \in \Sigma$ is called $(U_n)_n$-bounded if there exists $n \in *\mathbb{N}$ such that $A \subseteq U_n$ $\mu^\#$-almost everywhere.

**Theorem 2.1.2.** (Generalized Exhaustion theorem) Let $(Y_n)_n^\infty \subseteq \Sigma$ be a hyper infinite sequence satisfying $Y_n \uparrow X$ and $\mu^\#(Y_n) < *\infty$ for all $n \in *\mathbb{N}$.

Let $P$ be some property of $(Y_n)_n$-bounded

#-measurable sets, such that $A \in P$ iff $B \in P$ for all $B, \mu^\#(A \Delta B) = 0$. Suppose that any $(Y_n)_n$-bounded set $A$, $\mu^\#(A) > 0$, has a subset $B \in \Sigma, \mu^\#(B) > 0$ with the property $P$. Moreover, assume that either

(a) $A_1 \cup A_2 \in P$ for every $A_1, A_2 \in P$, or

(b) $\cup_{n \in *\mathbb{N}} B_n \in P$ for every at most hyper infinite family $(B_n)_n$ of pairwise disjoint sets possessing the property $P$.

Then there exists hyper infinite sequence $(X_n)_n^\infty \subseteq \Sigma$ such that $X_n \uparrow X$, and $P \ni X_n \subseteq Y_n$

for all $n \in *\mathbb{N}$. Moreover, there exists a pairwise disjoint sequence $(A_n)_n^\infty \subseteq \Sigma$ such that $\cup_{n \in *\mathbb{N}} A_n = X$ and $A_n \in P$ for all $n \in *\mathbb{N}$.

**Proof:** Let $A$ be a $(Y_n)_n$-bounded set with $\mu^\#(A) > 0$. Denote

$$P_A := \{B \in P : B \subseteq A\} \wedge m(A) := \sup\{\mu^\#(B) : B \in P_A\}.$$

**I(a)** Suppose $P$ satisfies (a). Then there exists a sequence $(F_n)_n^\infty \subseteq P_A$ such that $m(A) = \#\text{lim}_{n \to *\infty} \mu^\#(F_n)$, We may assume, that $F_n \uparrow$. By Proposition 2.5
the set $F = \bigcup_{n=1}^{\infty} F_n$ satisfies $\mu^#(F) = m(A)$. We show that $\mu^#(A) = m(A)$. If not
then $\mu^#(AF) > 0$. The set $AF$ has a subset of positive #-measure $F_0 \in P$.
Then $F_n \cup F_0 \in P_A$ and $\mu^#(F_n \cup F_0) > m(A)$ for a sufficiently large $n \in *\mathbb{N}$, which
contradicts to the definition of $m(A)$. Therefore, $\mu^#(A) = m(A)$.

Now we apply this for $A = Y_n$. Thus, there exists hyper infinite sequence $(X'_n)_n \subseteq \Sigma$
such that $X'_n \subseteq Y_n$, $X'_n$, $n \in P$, and $\mu^#(Y_n \Lambda X'_n) < n^{-1}$ for all $n \in *\mathbb{N}$. By (a), we may
assume that $X'_n \uparrow$. The set $X'_0 = \bigcup_{n=1}^{\infty} X'_n$ satisfies $Y_n \Lambda X'_0 \subseteq Y_n \Lambda X'_n$, so $\mu^#(Y_n \Lambda X'_0) < n^{-1}$
for all $n \in *\mathbb{N}$. Then $\mu^#(Y_n \Lambda X'_0) = 0$, and $\mu^#((\bigcup_{n=1}^{\infty} Y_n) \Lambda X'_0) = 0$, or $\mu^#(XX'_0) = 0$.
Let $X_n = (X'_n \cup (XX'_0) \cap Y_n)$, then the hyper infinite sequence $(X_n)_n$ has the required
properties. The desired pairwise disjoint sequence $(A_n)_{n=1}^{\infty}$ is given recurrently by
$A_1 = X_1$ and $A_{k+1} = X_{k+1} \setminus \bigcup_{i=1}^k A_i$.

I(b) Suppose $P$ satisfies (b). Let $F_A$ be the family of all pairwise disjoint families of elements of $P_A$ of nonzero measure. Then $F_A$ is ordered by inclusion and, obviously, satisfies the conditions of the Zorn lemma. Therefore, we have a maximal element in $F_A$, say $\Delta$. Then $\Delta$ is at most hyper infinite family, say $\Delta = \{D_n\}_n$. By (b), its union $D = \cup_n D_n$ is an element of $P_A$ as well. If $D$ is a proper subset of $A$, then $\mu^#(AD) > 0$. The set $AD$ has a subset $F \in P$ of the positive measure. Then $\Delta_1 := \Delta \cup \{F\}$ is an element of $F_A$ which is strictly greater then $\Delta$. The obtained contradiction, shows that $A \in P$ for every $(Y_n)_n$-bounded set $A$. So, we may take $X_n = Y_n$ for each $n \in *\mathbb{N}$.

Now we apply this for $A = Z_m = Y_m \setminus \bigcup_{k=1}^{m-1} Y_k$ be a pairwise
disjoint union, where $D_n^m \in P$ for all $n, m \in *\mathbb{N}$. The family $\{D_n^m\}_{n,m}$ is an at most
hyper infinite disjoint decomposition of $X$, say $\{D_n^m\}_{n,m} = (A_n)_{n=1}^{\infty}$. The sequence
$(A_n)_{n=1}^{\infty}$ satisfies the required properties.

**Theorem 2.1.3.** (The generalized Borel-Cantelli lemma) Let $(X, \Sigma, \mu^#)$ be a #-measure space. Assume that $\{A_n\}_n \subseteq \Sigma$ and $\text{Ext}-\sum_{n=1}^{\infty} \mu(A_n) < *\infty$ then $\lim_{n \to *\infty} \mu^#(A_n) = 0$.

## §2.2. Vector-valued #-measures

In this section, we extend the notion of a measure. Then we study the basic
operations with signed measures and present the Jordan decomposition theorem.

### 2.2.1. Vector-valued, signed and complex #-measures.

Let $\Sigma^#$ be a $\sigma^$-algebra of subsets of a set $X$, and let $E^# = (E^\# , \|\cdot\|_\#)$ be a non-Archimedean Banach space.

**Definition 2.2.1** A function $\mu^# : \Sigma^# \to E^# \cup \{ *\infty \}$ is called a vector-valued #-measure (or $E^$-valued measure) if

(a) $\mu^#(\emptyset) = 0$;

(b) $\mu^#(\bigcup_{k=1}^{\infty} A_k) = \text{Ext}-\sum_{k=1}^{\infty} \mu^#(A_k)$ for any pairwise disjoint sequence $(A_k)_k \subseteq \Sigma^#$;

(c) for any $A \in \Sigma^$, $\mu^#(A) = *\infty$, there exists $B \in \Sigma^$ such that $B \subseteq A$ and
$0 < \|\mu^#(B)\|_\# < *\infty$.

**Example 2.2.1** Take $\Sigma^# = P(*\mathbb{N})$, and $c_0^#$ is the non-Archimedean Banach space of all #-convergent $C_c^$-valued hyper infinite sequences with a fixed element $(\alpha_n)_n \in c_0^#$. Define now for any $A \subseteq *\mathbb{N}\psi(A) := (\beta_n)_n$, where $\beta_n = \alpha_n$ if $n \in A$ and $\beta_n = 0$ if $n \notin A$. Then $\psi$ is a $c_0^$-valued #-measure on $P(*\mathbb{N})$.
**Example 2.2.2** Let $X$ be a set and let $\Omega$ be a $\sigma^{\#}$-algebra in $P(X)$. Then for any family $\{\mu_k\}_{k=1}^m$ of finite #-measures on $\Omega$ and for any family $\{w_k\}_{k=1}^m$ of vectors of $\mathbb{R}_c^{\#n}$, the $\mathbb{R}_c^{\#n}$-valued #-measure $\Psi$ on $\Omega$ is defined by the formula
$$ \Psi(E) = \mathrm{Ext}-\sum_{k=1}^m \mu_k(E) \times w_k, (E \in \Omega). $$

**Example 2.2.3** Let $X$ be a set and let $\Omega$ be a $\sigma^{\#}$-algebra in $P(X)$. Then for any family $\{\mu_k\}_{k=1}^m$ of finite #-measures on $\Omega$, for any family $\{A_k\}_{k=1}^m$ of pairwise disjoint sets in $\Omega$, and for any family $\{w_k\}_{k=1}^m$ of $\mathbb{R}_c^{\#n}$, $n \in *\mathbb{N}$, the $\mathbb{R}_c^{\#n}$-valued #-measure $\Phi$ on $\Omega$ is defined by the formula
$$ \Phi(E) = \mathrm{Ext}-\sum_{k=1}^m \mu_k(E \cap A_k) \times w_k, (E \in \Omega). $$

## §3. The Lebesgue #-Integral

In the following consideration, we fix a $\sigma^{\#}$-finite #-measure space $(X, \mathcal{F}, \mu^{\#})$.

**Definition 3.1.** Let $A_i \in \mathcal{F}, i = 1, \dots, n \in *\mathbb{N}$, be such that $\mu^{\#}(A_i) < *\infty$ for all $i$, and $A_i \cap A_j = \emptyset$ for all $i \neq j$. The external function

$$ f(x) = \mathrm{Ext}-\sum_{i=1}^{n} \lambda_i \chi_{A_i}(x), \qquad (3.1) $$

$\lambda_i \in *\mathbb{R}_c^{\#}$, is called a simple external function. The Lebesgue external integral (Lebesgue #-integral) of a simple external function $f(x)$ is defined as

$$ \mathrm{Ext}-\int_X f(x)d^{\#}\mu^{\#} = \mathrm{Ext}-\sum_{i=1}^{n} \lambda_i \mu^{\#}(A_i). \qquad (3.2) $$

The Lebesgue external integral of a simple function is well defined.

**Notation 3.1.** Let $A_i \in \mathcal{F}, i = 1, \dots, n \in *\mathbb{N}$, be such that $\mu^{\#}(A_i) < *\infty$ for all $i$, and $A_i \cap A_j = \emptyset$ for all $i \neq j$. Let $f_1(x), f_2(x)$ be a simple external function such that

$$ (i) \quad 0 \le f_1(x) \le f_2(x) \text{ and } (ii) \quad f_1(x) = \mathrm{Ext}-\sum_{i=1}^{n} \lambda_{1,i} \chi_{A_i}(x), f_2(x) = \mathrm{Ext}-\sum_{i=1}^{n} \lambda_{2,i} \chi_{A_i}(x). $$

$$ \mathrm{Ext}-\sum_{i=1}^{n} \lambda_{1,i} \le \mathrm{Ext}-\sum_{i=1}^{n} \lambda_{2,i}, \qquad (3.3) $$

then we will write $f_1(x) \le s f_2(x)$.

**Definition 3.2.** Suppose that $\mu^{\#}$ is hyperfinite. Let $f: X \to *\mathbb{R}_c^{\#}$ be an arbitrary nonnegative bounded in $*\mathbb{R}_c^{\#}$ #-measurable external function and let $(f_n)_{n \in *\mathbb{N}}$, be a hyper infinite sequence of simple external functions which #-converges uniformly to $f$. Then the Lebesgue #-integral of $f$ is

$$ \mathrm{Ext}-\int_X f(x)d^{\#}\mu^{\#} = \#\lim_{n \to *\infty} \left( \mathrm{Ext}-\int_X f_n(x)d^{\#}\mu^{\#} \right). \qquad (3.4) $$

**Remark 3.1.** It can be easily shown that the #-limit in Definition 3.2 exists and does not depend on the choice of a hyper infinite sequence $(f_n)_{n \in *\mathbb{N}}$, and moreover, the hyper infinite sequence $(f_n)_{n \in *\mathbb{N}}$ can be chosen such that $0 \le f_n \le f$ for all $n \in *\mathbb{N}$.

**Notation 3.2.** Let $f_1: X \to *\mathbb{R}_c^{\#}$ and $f_2: X \to *\mathbb{R}_c^{\#}$ be an arbitrary nonnegative bounded in $*\mathbb{R}_c^{\#}$ #-measurable external functions and let $(f_{1,n})_{n \in *\mathbb{N}}$ and $(f_{2,n})_{n \in *\mathbb{N}}$ be a hyper infinite sequences of simple external functions which #-converges uniformly to $f_1$ and to $f_2$ correspondingly. We assume that for all $n \in *\mathbb{N}$ the inequality (3.3) is satisfied, then we will write $f_1(x) \le s f_2(x)$.

**Definition 3.3.** Let $f: X \to *\mathbb{R}_c^{\#}$ be a #-measurable function. Then the Lebesgue #-integral of $f$ is defined by
$$ \mathrm{Ext}-\int_X f(x)d^\#\mu^\# = \mathrm{Ext}-\int_X f^+(x)d^\#\mu^\# - \mathrm{Ext}-\int_X f^-(x)d^\#\mu^\#. \quad (3.5) $$

If both of these terms are finite or hyperfinite then the function $f$ is called #-integrable. In this case we write $f \in L_1^\# = L_1^\#(X, \mathcal{F}, \mu^\#)$.

**Notation 3.3.** We will use the following notation. For any $A \in \mathcal{F}$:

$$ \mathrm{Ext}-\int_A f(x)d^\#\mu^\# = \mathrm{Ext}-\int_X f(x)\chi_A(x)d^\#\mu^\#. \quad (3.6) $$

**Lemma 3.1.** (1) Let $f: X \to *\mathbb{R}_c^\#$ be an arbitrary nonnegative #-measurable function then

$$ \mathrm{Ext}-\int_X f(x)d^\#\mu^\# = \sup\left\{\mathrm{Ext}-\int_X \varphi(x)d^\#\mu^\# \mid \varphi \text{ is a simple function such that } 0 \le \varphi(x) \le s f(x)\right\}. \quad (3.7) $$

(2) If $f,g: X \to *\mathbb{R}_c^\#$ are #-measurable, $g$ is #-integrable, and $|f(x)|\le s g(x)$, then $f$ is #-integrable and

$$ \left| \mathrm{Ext}-\int_X f(x)d^\#\mu^\# \right| \le \mathrm{Ext}-\int_X g(x)d^\#\mu^\#. \quad (3.8) $$

(3) $\mathrm{Ext}-\int_X |f(x)|d^\#\mu^\# = 0$ if and only if $f(x) = 0$ #-a.e.

(4) If $f_1, f_2, \dots, f_n: X \to *\mathbb{R}_c^\#$, $n \in *\mathbb{N}$ are integrable then, for $\lambda_1, \lambda_2, \dots, \lambda_n \in *\mathbb{R}_c^\#$, the linear combination $\mathrm{Ext}-\sum_{i=1}^n \lambda_i f_i$ is #-integrable and

$$ \mathrm{Ext}-\int_X (\mathrm{Ext}-\sum_{i=1}^n \lambda_i f_i)d^\#\mu^\# = \mathrm{Ext}-\sum_{i=1}^n \left(\mathrm{Ext}-\int_X \lambda_i f_i d^\#\mu^\#\right). \quad (3.9) $$

(5) Let $f \in L_1^\#(X, \mathcal{F}, \mu^\#)$, then the formula

$$ v^\#(A) = \mathrm{Ext}-\int_A f(x)d^\#\mu^\# = \mathrm{Ext}-\int_X f(x)\chi_A(x)d^\#\mu^\# \quad (3.10) $$

defines a signed #-measure on the $\sigma^\#$-algebra $\mathcal{F}$.

**Remark 3.2.** Assume that $f,g: X \to *\mathbb{R}_c^\#$ are #-integrable functions and such that $0 \le f \le s g$ #-a.e., then

$$ \mathrm{Ext}-\int_X f(x)d^\#\mu^\# \le \mathrm{Ext}-\int_X g(x)d^\#\mu^\#. $$

#-Convergence theorem

**Definition 3.4.** We say that a hyper infinite sequence $\{f_n\}_{n=1}^\infty$ of #-integrable functions $L_1^#$-#-converges to $f$ (or #-converges in $L_1^\#(X, \mathcal{F}, \mu^\#)$) if

$$ \mathrm{Ext}-\int_X |f_n - f|d^\#\mu^\# \to_\# 0 \quad \text{as } n \to_* \infty. \quad (3.11) $$

**Theorem 3.1** (The monotone #-convergence theorem) If $\{f_n\}_{n=1}^\infty$ is a hyper infinite sequence in $L_1^+(X, \mathcal{F}, \mu^\#)$ such that $f_j \le s f_{j+1}$ for all $j$ and $f(x) = \sup_{n \in *\mathbb{N}} f_n(x)$ then

$$ \mathrm{Ext}-\int_X f(x)d^\#\mu^\# = \#\mathrm{-lim}_{n \to *\infty} \mathrm{Ext}-\int_X f_n(x)d^\#\mu^\#. \quad (3.12) $$

**Proof:** The #-limit of the increasing sequence

$$ \left( \mathrm{Ext}-\int_X f_n(x)d^\#\mu^\# \right)_{n=1}^\infty $$

(*-finite or *-infinite) exists. Moreover by (3.2),
$$Ext-\int_X f_n(x)d^\#\mu^\# \le Ext-\int_X f(x)d^\#\mu^\#$$

for all $n \in *\mathbb{N}$, so

$$\#\lim_{n \to *\infty} \left( Ext-\int_X f_n(x)d^\#\mu^\# \right) \le Ext-\int_X f(x)d^\#\mu^\#.$$

To establish the reverse inequality, fix $\alpha \in (0,1)$, let $\varphi$ be a simple function with $0 \le \varphi \le f$, and let $E_n = \{x : f_n(x) \ge \alpha\varphi(x)\}$. Then $(E_n)_{n=1}^{*\infty}$ is an increasing hyper infinite sequence of #-measurable sets whose union is X, and we have

$$Ext-\int_X f_n(x)d^\#\mu^\# \ge Ext-\int_{E_n} f_n(x)d^\#\mu^\# \ge \alpha \left( Ext-\int_{E_n} \varphi(x)d^\#\mu^\# \right) \quad (3.13)$$

By (3.10) and by Proposition 2.5,

$$\#\lim_{n \to *\infty} \left( Ext-\int_{E_n} \varphi(x)d^\#\mu^\# \right) = Ext-\int_X \varphi(x)d^\#\mu^\#, \quad (3.14)$$

and hence

$$\#\lim_{n \to *\infty} \left( Ext-\int_{E_n} f_n(x)d^\#\mu^\# \right) \ge \alpha \left( Ext-\int_X \varphi(x)d^\#\mu^\# \right). \quad (3.15)$$

Since this is true for all $\alpha, 0 < \alpha < 1$, it remains true for $\alpha = 1$:

$$\#\lim_{n \to *\infty} \left( Ext-\int_{E_n} f_n(x)d^\#\mu^\# \right) \ge Ext-\int_X \varphi(x)d^\#\mu^\#. \quad (3.16)$$

Using Lemma 3.1.(1), we may take the supremum over all simple functions $\varphi$, $0 \le \varphi \le s$. Thus

$$\#\lim_{n \to *\infty} \left( Ext-\int_{E_n} f_n(x)d^\#\mu^\# \right) \ge Ext-\int_X f(x)d^\#\mu^\#. \quad (3.17)$$

Proofs of the following two corollaries of Theorem 3.1 are straightforward.

**Corollary 3.1** If $(f_n)_{n=1}^{*\infty}$ is a hyper infinite sequence in $L_+^1(X)$ and $f = Ext-\sum_{n=1}^{*\infty} f_n$ pointwise then

$$Ext-\int_X f(x)d^\#\mu^\# = Ext-\sum_{n=1}^{*\infty} \left( Ext-\int_X f_n(x)d^\#\mu^\# \right). \quad (3.18)$$

**Corollary 3.2** If $(f_n)_{n=1}^{*\infty}$ is a hyper infinite sequence in $L_+^1(X)$, $f \in L_+^1(X)$, and $f_n \to_s f$ $\mu^\#$-a.e., then

$$Ext-\int_X f_n(x)d^\#\mu^\# \to_s Ext-\int_X f(x)d^\#\mu^\#. \quad (3.19)$$

**Theorem 3.2** (Generalized Fatou's lemma) If $(f_n)_{n=1}^{*\infty}$ is any hyper infinite sequence in $L_+^1(X)$ then

$$Ext-\int_X \#\liminf_{n \to *\infty} (f_n(x))d^\#\mu^\# \le \#\liminf_{n \to *\infty} \left( Ext-\int_X f_n(x)d^\#\mu^\# \right). \quad (3.20)$$

**Theorem 3.3** (The dominated #-convergence theorem) Let $f$ and $g$ be #-measurable, let $f_n$ be #-measurable for any $n \in *\mathbb{N}$ such that $|f_n(x)| \le s$ $g(x)$ #-a.e., and $f_n \to_s f$ #-a.e. If $g$ is #-integrable then $f$ and $f_n$ are also #-integrable and

$$Ext-\int_X f(x)d^\#\mu^\# = \#\lim_{n \to *\infty} Ext-\int_X f_n(x)d^\#\mu^\#. \quad (3.21)$$

**Proof:** $f$ is #-measurable and, since $|f| \le s$ $g$ $\mu^\#$-a.e., we have $f \in L_+^1(X)$. We have that $g+f_n \ge 0$ $\mu^\#$-a.e. and $g-f_n \ge 0$ so, by Fatou's lemma,
$$
\begin{align}
& \mathrm{Ext}-\int_X g d^\# \mu^\# + \mathrm{Ext}-\int_X f d^\# \mu^\# \le \#\text{-lim } \inf_{n \to \infty} \left( \mathrm{Ext}-\int_X [g+f_n] d^\# \mu^\# \right) = \nonumber \\
& \qquad \mathrm{Ext}-\int_X g d^\# \mu^\# + \#\text{-lim } \inf_{n \to \infty} \left( \mathrm{Ext}-\int_X f_n d^\# \mu^\# \right), \tag{3.22} \\
& \mathrm{Ext}-\int_X g d^\# \mu^\# - \mathrm{Ext}-\int_X f d^\# \mu^\# \le \#\text{-lim } \inf_{n \to \infty} \left( \mathrm{Ext}-\int_X [g-f_n] d^\# \mu^\# \right) = \nonumber \\
& \qquad = \mathrm{Ext}-\int_X g d^\# \mu^\# - \#\text{-lim } \sup_{n \to \infty} \left( \mathrm{Ext}-\int_X f_n d^\# \mu^\# \right) \nonumber
\end{align}
$$

Therefore

$$
\#\text{-lim } \inf_{n \to \infty} \left( \text{Ext}-\int_X f_n d^\# \mu^\# \right) \ge \text{Ext}-\int_X f d^\# \mu^\# \ge \#\text{-lim } \sup_{n \to \infty} \left( \text{Ext}-\int_X f_n d^\# \mu^\# \right) \quad (3.23)
$$

and the required result follows from (3.23).

§ 4. #-Convergence in #-measure.

**Definition 4.1.** We say that a hyper infinite sequence $(f_n)_{n=1}^{\infty}$ of #-measurable functions on $(X, M, \mu^#)$ is Cauchy in #-measure if, for every $\varepsilon \approx 0, \varepsilon > 0$,

$$
\mu^{\#}(\{x : |f_n(x) - f_m(x)| \ge \varepsilon\}) \to_{\#} 0 \text{ as } m, n \to^{*\infty}, \quad (4.1)
$$

and that $(f_n)_{n=1}^{\infty}$ #-converges in #-measure to $f$ if, for every $\varepsilon \approx 0, \varepsilon > 0$,

$$
\mu^{\#}(\{x : |f_n(x) - f(x)| \ge \varepsilon\}) \to_{\#} 0 \text{ as } n \to^{*\infty}. \quad (4.2)
$$

**Proposition 4.1.** If $f_n \to_{\#} f$ in $L^1$ then $f_n \to_{\#} f$ in #-measure.

**Proof.** Let $E_{n,\varepsilon} = \{x : |f_n(x) - f(x)| \ge \varepsilon\}$. Then

$$
\mathrm{Ext}-\int_X |f_n - f| d\mu^\# \ge \mathrm{Ext}-\int_{E_{n,\varepsilon}} |f_n - f| d\mu^\# \ge \varepsilon \mu^\#(E_{n,\varepsilon}),
$$

$$
\text{so } \mu(E_{n,\varepsilon}) \le \varepsilon^{-1} \mathrm{Ext}-\int_X |f_n - f| d\mu^\# \to_{\#} 0.
$$

**Theorem 3.1.** Suppose that $(f_n)_{n=1}^{\infty}$ is Cauchy in #-measure. Then there is a
\#-measurable function $f$ such that $f_n \to_{\#} f$ in \#-measure, and there is a
hyper infinite subsequence $(f_{n_j})_{j \in \mathbb{N}}$ that \#-converges to $f$ \#-a.e. Moreover, if
$f_n \to_{\#} g$ in \#-measure then $g = f$ \#-a.e.

**Proof.** We can choose a hyper infinite subsequence $(g_j)_{j} = (f_{n_j})_j$ of $(f_n)_{n=1}^{\infty}$ such that if $E_j = \{x : |g_j(x) - g_{j+1}(x)| \ge 2^{-j}\}$ then $\mu^\#(E_j) \le 2^{-j}$. If $F_k = \bigcup_{j=k}^{\infty} E_j$ then

$$
\mu^{\#}(F_k) \le \mathrm{Ext}-\sum_{j=k}^{\infty} 2^{-j} = 2^{1-k}, \text{ and if } x \notin F_k \text{ we have for } i \ge j \ge k
$$

$$
|g_j(x) - g_i(x)| \le \mathrm{Ext}-\sum_{l=j}^{i-1} |g_{l+1}(x) - g_l(x)| \le \mathrm{Ext}-\sum_{l=j}^{i-1} 2^{1-j}. \quad (4.3)
$$

Thus $(g_j)_j$ is pointwise Cauchy on $F_k^c$. Let $F = \bigcap_{k=1}^{\infty} F_k = \lim \sup_j E_j$. Then $\mu^\#(F) = 0$,
and if we set $f(x) = \lim_{j \to \infty} g_j(x)$ for $x \notin F$, and $f(x) = 0$ for $x \in F$, then $f$ is
\#-measurable and $g_j \to \# f$ a.e. By (4.3), we have that $|g_j(x) - f(x)| \le 2^{1-j}$ for $x \notin F_k$
and $j \ge k$. Since $\mu^\#(F_k) \to \# 0$ as $k \to \infty$, it follows that $g_j \to \# f$ in \#-measure,
because

$$
\{x : |f_n(x) - f(x)| \ge \varepsilon\} \subseteq \{x : |f_n(x) - g_j(x)| \ge (1/2)\varepsilon\} \cup \{x : |g_j(x) - f(x)| \ge (1/2)\varepsilon\}, \quad (4.4)
$$

and the sets on the right both have infinite small #-measure when $n$ and $j$ are infinite
large. Likewise, if $f_n \to_g g$ in #-measure

$$ \{x : |f(x) - g(x)| \ge \varepsilon\} \subseteq \{x : |f(x) - f_n(x)| \ge (1/2)\varepsilon\} \cup \{x : |f_n(x) - g(x)| \ge (1/2)\varepsilon\} \quad (4.5) $$

for all $n \in *\mathbb{N}$, hence $\mu^{\#}(\{x : |f(x) - g(x)| \ge \varepsilon\}) = 0$ for all $\varepsilon > 0$, and $f = g$ #-a.e.

**Theorem 3.2** Let $f_n \to_g f$ in $L_1^\#$ then there is a hyper infinite subsequence $(f_{n_k})_k$ such that $f^{n_k} \to_g f$ #-a.e.

**Proof.** Let $E_n, \varepsilon = \{x : |f_n(x) - f(x)| \ge \varepsilon\}$. Then

$$ \mathrm{Ext} - \int_X |f_n - f| d^\# \mu \ge \mathrm{Ext} - \int_{E_{n,\varepsilon}} |f_n - f| d^\# \mu \ge \varepsilon \mu^\#(E_{n,\varepsilon}), $$

so $\mu^\#(E_{n,\varepsilon}) \to_\# 0$. Then, by Theorem 3.1, there is a hyper infinite subsequence $(f_{n_k})_k$ such that $f_{n_k} \to_\# f$ #-a.e.

# 5. The Extension of #-Measure

## § 5.1. Outer #-measures.

**Definition 5.1.1.** Let $X$ be a nonempty set. An outer #-measure

(or #-submeasure) on $X$ is a function $\zeta^\# : \tilde{P}(X) \to [0, *^\infty], \tilde{P}(X) \subset P(X)$ that satisfies:

(a) $\zeta^\#(\emptyset) = 0$;

(b) $\zeta^\#(A) \le \zeta(B)$ if $A \subseteq B$;

(c) $\zeta^\#(\bigcup_{j=1}^{*\infty} A_j) \le \mathrm{Ext} - \sum_{j=1}^{*\infty} \zeta^\#(A_j)$ for all hyper infinite sequences $(A_j)_{j=1}^{*\infty}$ in $\tilde{P}(X)$.

The common way to obtain an outer #-measure is to start with a family $G$ of “elementary sets” on which a notion of measure is defined (such as rectangles or cubes in $*\mathbb{R}_c^{*\infty}$ and then approximate arbitrary sets from the outside by hyper infinite unions of members of $G$.

**Proposition 5.1.1** Let $G \subseteq \tilde{P}(X)$ be a set such that $\emptyset \in G, X \in G$ and let $\rho : G \to [0, *^\infty]$ be a function such that $\rho(\emptyset) = 0$. For any $A \subseteq X$, define

$$ \zeta^\#(A) = \rho^*(A) = \inf \left\{ \mathrm{Ext} - \sum_{j=1}^{*\infty} \rho(G_j) : G_j \in G \text{ and } A \subseteq \bigcup_{j=1}^{*\infty} G_j \right\}. \quad (5.1.1) $$

if $\rho^*(A)$ exists. Then $\zeta^\#$ is an outer #-measure.

**Definition 5.1.2.** We will say that $A \subseteq X$ is admissible if $\rho^*(A)$ exists.

**Proof.** For any admissible $A \subseteq X$, $\zeta^\#(A)$ is well defined. Obviously $\zeta^\#(\emptyset) = 0$. To prove $*$-countable subadditivity, suppose $\{A_j\}_{j=1}^{*\infty} \subseteq \tilde{P}(X)$ and $\varepsilon \approx, \varepsilon > 0$.

For each $j \in *\mathbb{N}$, there exists $\{G_j^j\}_{j=1}^{*\infty} \subseteq G$ such that $A_j \subseteq \bigcup_{k=1}^{*\infty} G_k^j$ and

$$ \mathrm{Ext} - \sum_{k=1}^{*\infty} \rho(G_k^j) \le \zeta^\#(A_j) + \varepsilon 2^{-j}. \text{ Then if } A = \bigcup_{j=1}^{*\infty} A_j, \text{ we have } A \subseteq \bigcup_{j,k=1}^{*\infty} G_k^j \text{ and} $$

$$ \mathrm{Ext} - \sum_{j,k=1}^{*\infty} \rho(G_k^j) \le \sum_{j=1}^{*\infty} \zeta(A_j) + \varepsilon, \text{ whence } \zeta^\#(A) \le \mathrm{Ext} - \sum_{j=1}^{*\infty} \zeta^\#(A_j) + \varepsilon. \text{ Since } \varepsilon > 0 \text{ is arbitrary, we have done.} $$

**Definition 5.1.3.** A set $A \subseteq X$ is called $\zeta^\#$-measurable if $\rho^*(A)$ exists and $\forall B \subseteq X$ such that $\rho^*(B)$ exists the equality (5.1.2) holds

$$ \zeta^\#(B) = \zeta^\#(B \cap A) + \zeta^\#(B \cap (X \setminus A)). \quad (5.1.2) $$
Of course, the inequality $\zeta^{\#}(B) \le \zeta^{\#}(B \cap A) + \zeta^{\#}(B \cap (X\backslash A))$ holds for any (admissible) set $A$ and $B$.

So, to prove that $A$ is $\zeta^{\#}$-measurable, it suffices to prove the reverse inequality, which is trivial if $\zeta^{\#}(B) = *_{\infty}$. Thus, we see that $A$ is $\zeta^{\#}$-measurable iff for any admissible $B \subseteq X$, $\zeta^{\#}(B) < *_{\infty}$

$$ \zeta^{\#}(B) \ge \zeta^{\#}(B \cap A) + \zeta^{\#}(B \cap (X\backslash A)). \quad (5.1.3) $$

**Theorem 5.1.1 (Generalized Caratheodory's theorem)** Let $\zeta^{\#}$ be an outer #-measure on $X$. Then the family $\Sigma$ of all $\zeta^{\#}$-measurable sets is a $\sigma^{\#}$-algebra, and the restriction of $\zeta^{\#}$ to $\Sigma$ is a complete #-measure.

**Proof:** First, we observe that $\Sigma$ is closed under complements, since the definition of $\zeta^{\#}$-measurability of $A$ is symmetric in $A$ and $A^c \triangleq X\backslash A$. Next, if $A, B \in \Sigma$ and $E \subseteq X$,

$$ \zeta^{\#}(E) = \zeta^{\#}(E \cap A) + \zeta^{\#}(E \cap A^c) = \zeta^{\#}(E \cap A \cap B) + \zeta^{\#}(E \cap A \cap B^c) + \zeta^{\#}(E \cap A^c \cap B) + \\ + \zeta^{\#}(E \cap A^c \cap B^c). $$

But $(A \cup B) = (A \cap B) \cup (A \cap B^c) \cup (A^c \cap B)$ so, by subadditivity, $\zeta^{\#}(E \cap A \cap B) + \zeta^{\#}(E \cap A \cap B^c) + \zeta^{\#}(E \cap A^c \cap B) \ge \zeta^{\#}(E \cap (A \cup B))$, and hence $\zeta^{\#}(E) \ge \zeta^{\#}(E \cap (A \cup B)) + \zeta^{\#}(E \cap (A \cup B)^c)$.

It follows that $A \cup B \in \Sigma$, so $\Sigma$ is an algebra. Moreover, if $A, B \in \Sigma$ and $A \cap B = \emptyset$, $\zeta^{\#}(A \cup B) = \zeta^{\#}((A \cup B) \cap A) + \zeta^{\#}((A \cup B) \cap A^c) = \zeta^{\#}(A) + \zeta^{\#}(B)$, so $\zeta^{\#}$ is hyperfinitely additive on $\Sigma$.

To show that $\Sigma$ is a $\sigma^{\#}$-algebra, it suffices to show that $\Sigma$ is closed under $*_{\infty}$-countable disjoint unions. If $(A_j)_{j=1}^{*_{\infty}}$ is a sequence of disjoint sets in $\Sigma$, set

$$ B_n = \bigcup_{j=1}^{n} A_j \cap B = \bigcup_{j=1}^{*_{\infty}} A_j. \text{ Then, for any admissible } E \subseteq X, $$

$$ \zeta^{\#}(E \cap B_n) = \zeta^{\#}(E \cap B_n \cap A_n) + \zeta^{\#}(E \cap B_n \cap A_n^c) = \zeta^{\#}(E \cap A_n) + \zeta^{\#}(E \cap B_{n-1}), $$

so a hyperfinite induction shows that $\zeta^{\#}(E \cap B_n) = \text{Ext}-\sum_{j=1}^{n} \zeta^{\#}(E \cap A_j)$. Therefore

$$ \zeta^{\#}(E) = \zeta^{\#}(E \cap B_n) + \zeta^{\#}(E \cap B_n^c) \ge \text{Ext}-\sum_{j=1}^{n} \zeta^{\#}(E \cap A_j) + \zeta^{\#}(E \cap B^c) $$

and, letting $n \to *_{\infty}$, we obtain

$$ \zeta^{\#}(E) \ge \text{Ext}-\sum_{j=1}^{*_{\infty}} \zeta^{\#}(E \cap A_j) + \zeta^{\#}(E \cap B^c) \ge \zeta^{\#}\left(\bigcup_{j=1}^{*_{\infty}} E \cap A_j\right) + \zeta^{\#}(E \cap B^c) = \zeta^{\#}(E \cap B) + \\ + \zeta^{\#}(E \cap B^c) \ge \zeta^{\#}(E). $$

Thus the inequalities in this last calculation become equalities. It follows $B \in \Sigma$.

Taking $E = B$ we have $\zeta^{\#}(B) = \text{Ext}-\sum_{j=1}^{*_{\infty}} \zeta^{\#}(A_j)$, so $\zeta^{\#}$ is $\sigma^{\#}$-additive on $\Sigma$. Finally, if

$\zeta^{\#}(A) = 0$ then we have for any admissible set $E \subseteq X$

$$ \zeta^{\#}(E) \le \zeta^{\#}(E \cap A) + \zeta^{\#}(E \cap A^c) = \zeta^{\#}(E \cap A^c) \le \zeta^{\#}(E), \text{ so } A \in \Sigma. $$

Therefore $\zeta^{\#}(E \cap A) = 0$ and $\zeta^{\#}|_{\Sigma}$ is a complete #-measure.

Combination of Proposition 5.1.1 and Theorem 5.1.1 gives the following corollary which is also called generalized Caratheodory's theorem.

**Corollary 5.1.1** Let $G \subseteq P(X)$ be a set such that $\emptyset \in G, X \in G$, and let $\rho: G \to [0, *_{\infty}]$ satisfy $\rho(\emptyset) = 0$. Then the family $\Sigma$ of all $\rho^*$-measurable sets (where $\rho^*$ is given by (5.1.1)) is a $\sigma^{\#}$-algebra, and the restriction $\rho^*|_{\Sigma}$ of $\rho^*$ to $\Sigma$ is a complete #-measure.
**Definition 5.1.4** Let $\bar{A}$ be an algebra of subsets of $X$, i.e. $\bar{A}$ contains $\emptyset$ and $X$, and $\bar{A}$ is closed under hyperfinite intersections and complements. A function

$\zeta: \bar{A} \to [0, *^\infty]$ is called a \#-premeasure if $\zeta(\emptyset) = 0$ and $\zeta(\bigcup_{j=1}^{*^\infty} A_j) = \text{Ext}-\sum_{j=1}^{*^\infty} \zeta(A_j)$ for any disjoint sequence $(A_j)_{j \in *^\mathbb{N}}$ of elements of $\bar{A}$ such that $\bigcup_{j=1}^{*^\infty} A_j \in \bar{A}$.

**Theorem 5.1.2** If $\zeta$ is a \#-premeasure on an algebra $\bar{A} \subseteq P(X)$ and $\zeta^*: P(X) \to [0, *^\infty]$ is given by (5.1.1) then $\zeta^*|_A = \zeta$ and every $A \in \bar{A}$ is \#-measurable.

## § 5.2. The Lebesgue and Lebesgue – Stieltjes \#-measure on $\mathbb{R}^\#_c$.

The most important application of generalized Caratheodory's theorem is the construction of the Lebesgue \#-measure on $\mathbb{R}^\#_c$. Take $G$ as the set of all intervals $[a, b]$, where $a, b \in \mathbb{R}^\#_c \cup \{-*^\infty, +*^\infty\}$ and $[a, b] = \emptyset$ if $a > b$. Define the function $\rho: G \to \mathbb{R}^\#_c \cup \{*^\infty\}$ by

$$ \forall a \forall b (a \le b) [\rho([a, b]) = b - a] \text{ and } \forall a \forall b (a > b) [\rho([a, b]) = 0]. \quad (5.2.1) $$

The function $\rho$ has the obvious extension (which we denote also by $\rho$) to the algebra $A$ generated by all intervals, and this extension is a \#-premeasure on $A$. The $\sigma^\#$-algebra $\Sigma$ given by Corollary 5.1.1 is called the the Lebesgue $\sigma^\#$-algebra in $\mathbb{R}$, and the restriction of $\rho^*$ to $\Sigma = \Sigma(*\mathbb{R}^\#_c)$ is called the Lebesgue \#-measure on $*\mathbb{R}^\#_c$ and is denoted by $\mu^\#$. By Theorem 5.1.2, $\mu^\#$ is the unique extension of $\rho$. By the construction, $B^\#(*\mathbb{R}^\#_c) \subseteq \Sigma(*\mathbb{R}^\#_c)$. Hence the Lebesgue \#-measure is a Borel \#-measure. It can be shown that $B^\#(*\mathbb{R}^\#_c) \neq *\mathbb{R}^\#_c$ and that the Lebesgue \#-measure can be obtained also as the completion of any Borel \#-measure $\omega^\#$ such that $\omega^\#([a, b]) = b - a(\forall a \le b)$.

The notion of the Lebesgue measure on $*\mathbb{R}^\#_c$ has the following generalization. Suppose that $\mu^\#$ is a $\sigma^\#$-finite Borel measure on $*\mathbb{R}^\#_c$, and let $\forall x \in *\mathbb{R}^\#_c$

$$ F(x) = \mu^\#((-*^\infty, x]) \quad (5.2.2) $$

Then $F$ is increasing and right \#-continuous. Moreover, if $b > a$, $(-*^\infty, b] = (-*^\infty, a] \cup (a, b]$, so $\mu^\#((a, b]) = F(b) - F(a)$.

Our procedure used above can be to turn this process around and construct a measure $\mu$ starting from an increasing, right-continuous function $F$. The special case $F(x) = x$ will yield the usual Lebesgue \#-measure. As building blocks we can use the left-\#-open, right-\#-closed intervals in $*\mathbb{R}^\#_c$ i.e. sets of the form $(a, b]$ or $(a, *^\infty)$ or $\emptyset$, where $-*^\infty \le a < b < *^\infty$. We call such sets $h$-intervals. The family $A$ of all finite disjoint unions of $h$-intervals is an algebra, moreover, the $\sigma^\#$-algebra generated by $A$ is the \#-Borel algebra $B^\#(*\mathbb{R}^\#_c)$.

**Lemma 5.2.1.** Given an increasing and right \#-continuous function $F: *\mathbb{R}^\#_c \to *\mathbb{R}^\#_c$, if $(a_j, b_j](j = 1, \dots, n), n \in *\mathbb{N}$ are disjoint $h$-intervals, let

$$ \mu_0^\#\left(\bigcup_{j=1}^n (a_j, b_j]\right) = \text{Ext}-\sum_{j=1}^n [F(b_j) - F(a_j)], \quad (5.2.3) $$

and let $\mu_0^\#(\emptyset) = 0$. Then $\mu_0^\#$ is a \#-premeasure.

**Lemma 5.2.2.**f Assume that $\{(a_\alpha, b_\alpha) | \alpha \in G\}$ is a hyperfinite or \*-countable
family of intervals in $\mathbb{R}_c^\#$ such that $[0, 1] \subseteq \cup_{a \in G} (a_a, b_a)$ then $\mathrm{Ext}-\sum_{a \in G} |a_a - b_a| > 1$.

**Theorem 5.2.1** If $F: \mathbb{R}_c^\# \to \mathbb{R}_c^\#$ is any increasing, right #-continuous function, there is a unique Borel #-measure $\mu_F^\#$ on $\mathbb{R}_c^\#$ such that $\forall a \forall b (a, b \in \mathbb{R}_c^\#)$

$$ \mu_F^\#((a, b]) = F(b) - F(a). $$

If $G$ is another such function then $\mu_F^\# = \mu_G^\#$ iff $F - G$ is constant.

Conversely, if $\mu^\#$ is a Borel #-measure on $\mathbb{R}_c^\#$ that is gyperfinite on all #-bounded #-Borel sets, and we define $F(x) = \mu^\#((0, x])$ if $x > 0$, $F(x) = 0$ if $x = 0$, $F(x) = -\mu^\#((x, 0])$ if $x < 0$,

then $F$ is increasing and right #-continuous function, and $\mu^\# = \mu_F^\#$.

**Proof:** Each $F$ induces a #-premeasure on $B^\#(\mathbb{R}_c^\#)$ by Lemma 5.1.1. It is clear that $F$ and $G$ induce the same #-premeasure iff $F - G$ is constant, and that these #-premeasures are $\sigma^\#$-finite (since $\mathbb{R}_c^\# = \bigcup_{j=-\infty}^{\infty} (j, j+1]$). The first two assertions

follow now from Lemma 5.2.2. As for the last one, the monotonicity of $\mu^\#$ implies the monotonicity of $F$, and the #-continuity of $\mu^\#$ from above and from below implies the right #-continuity of $F$ for $x \ge 0$ and $x < 0$. It is evident that $\mu^\# = \mu_F^\#$ on algebra $A$, and hence $\mu^\# = \mu_F^\#$ on $B^\#(\mathbb{R}_c^\#)$ (accordingly to Lemma 5.2.4). Lebesgue – Stieltjes #-measures possess some important and useful regularity properties.

Let us fix a complete Lebesgue – Stieltjes #-measure $\mu^\#$ on $\mathbb{R}_c^\#$ associated to an increasing, right #-continuous function $F$. We denote by $\Sigma_{\mu^\#}$ the Lebesgue algebra correspondent to $\mu^\#$. Thus, for any $E \in \Sigma_{\mu^\#}$,

$$ \begin{aligned} \mu^\#(E) &= \inf \left\{ \mathrm{Ext} - \sum_{j=1}^{\infty} [F(b_j) - F(a_j)] \middle| E \subseteq \bigcup_{j=1}^{\infty} (a_j, b_j] \right\} = \\ &= \inf \left\{ \mathrm{Ext} - \sum_{j=1}^{\infty} \mu_F^\#((a_j, b_j]) \middle| E \subseteq \bigcup_{j=1}^{\infty} (a_j, b_j] \right\} \end{aligned} \quad (5.2.4) $$

if infinum in RHS of (5.2.4) exists. Since $B^\#(\mathbb{R}_c^\#) \subseteq \Sigma_{\mu^\#}$, we may replace in the second formula for $\mu^\#(E)$ h-intervals by #-open intervals, namely

**Lemma 5.2.3** For any $E \in \Sigma_{\mu^\#}$,

$$ \mu^\#(E) = \inf \left\{ \mathrm{Ext} - \sum_{j=1}^{\infty} \mu_F^\#((a_j, b_j)) \middle| E \subseteq \bigcup_{j=1}^{\infty} (a_j, b_j) \right\}. \quad (5.2.5) $$

**Theorem 5.2.2** If $E \in \Sigma_{\mu^\#}$ then

$$ \begin{aligned} E \in \Sigma_{\mu^\#} &= \inf\{\mu^\#(U) : U \supseteq E \text{ and } U \text{ is #-open}\} = \\ &= \sup\{\mu^\#(K) : K \subseteq E \text{ and } K \text{ is #-compact}\}. \end{aligned} \quad (5.2.6) $$

**Proof.** By Lemma 5.2.2, for any $\varepsilon \approx, \varepsilon > 0$, there exist intervals $(a_j, b_j)$ such that $E \subseteq \bigcup_{j=1}^{\infty} (a_j, b_j)$ and $\mu^\#(E) \le \mathrm{Ext} - \sum_{j=1}^{\infty} \mu^\#((a_j, b_j)) + \varepsilon$. If $U = \bigcup_{j=1}^{\infty} (a_j, b_j)$ then $U$ is #-open, $E \subseteq U$, and $\mu^\#(U) \le \mu^\#(E) + \varepsilon$. On the other hand, $\mu^\#(U) \ge \mu^\#(E)$ whenever $E \subseteq U$ so the first equality is valid.

For the second one, suppose first that $E$ is bounded in $\mathbb{R}_c^\#$. If $E$ is #-closed then $E$ is #-compact and the equality is obvious. Otherwise, given $\varepsilon \approx, \varepsilon > 0$, we can
choose an #-open U, (#-E)\E ⊆ U, such that µ#(U) ≤ µ#((#-E)\E) + ε.

Let K = (#-E) \U. Then K is #-compact, K ⊆ E, and

$$
\begin{align*}
\mu^\#(K) &= \mu^\#(E) - \mu^\#(E \cap U) = \mu^\#(E) - [\mu^\#(U) - \mu^\#(U\E)] \ge \\
&\ge \mu^\#(E) - \mu^\#(U) + \mu^\#((#-E)\E) \ge \mu^\#(E) - \varepsilon.
\end{align*}
$$

If E is unbounded in *ℝ<sup>#</sup><sub>c</sub>, let E<sub>j</sub> = E ∩ (j, j + 1]. By the preceding argument, for
any ε ≈, ε > 0, there exist a #-compact K<sub>j</sub> ⊆ E<sub>j</sub> with μ<sup>#</sup>(K<sub>j</sub>) ≥ μ<sup>#</sup>(E<sub>j</sub>) - ε2<sup>-j</sup>. Let

$$
H_n = \bigcup_{j=n}^{j=n} K_j. \text{ Then } H_n \text{ is } \#\text{-compact, } H_n \subseteq E, \text{ and } \mu^\#(H_n) \ge \mu^\# \bigcup_{j=n}^{j=n} (E_j) - \varepsilon.
$$

Since $\mu^\#(E) = \#\text{-lim}_{n \to *\infty} \mu^\#\left(\bigcup_{j=-n}^{j=n} E_j\right)$, the result follows.

**Theorem 5.2.3.** If $E \subseteq *\mathbb{R}_c^#$, the following are equivalent:

(a) $E \in \Sigma_{\mu}^{\#}$;

(b) $E = VW_1$, where V is a $G_{\delta}^-$-set and $\mu^\#(N_1) = 0$;

(c) $E = H \cup N_2$, where $H$ is an $F_{\sigma}^-$-set and $\mu^\#(N_2) = 0$.

**Theorem 5.2.4.** If $E \in \Sigma_{\mu^\#}$ and $\mu^\#(E) < *\infty$ then, for every $\varepsilon \approx, \varepsilon > 0$, there is a set $A$ that is a hyperfinite union of #-open intervals such that $\mu^\#(E\Delta A) < \varepsilon$.

**Lemma 5.2.4** Let $A \subseteq P(X)$ be an algebra, let $\mu_0^\#$ be a $\sigma^\#$-finite #-premeasure on $A$, and let $\Omega$ be the $\sigma^\#$-algebra generated by $A$. Then there exists a unique extension of $\mu_0^\#$ to a #-measure $\mu^\#$ on $\Omega$.

§ 5.3. Product #-measures.

**Definition 5.3.1.** Let {(*X*<sub>*a*</sub>, *F*<sub>*a*</sub>, *μ*<sub>*a*</sub><sup>#</sup>)}<sub>*a*∈Δ</sub> be a nonempty family of #-measure spaces. We define the family Ω of blocks:

$$
\begin{align}
A(A_{a_1}, A_{a_2}, \dots, A_{a_n}) &:= \nonumber \\
&= A_{a_1} \times A_{a_2} \times \dots \times A_{a_n} \times \mathrm{Ext-} \prod_{\substack{a \in \Delta, a \neq a_k, 1 \le k \le n \\ a \neq a_k}} X_a, \tag{5.3.1}
\end{align}
$$

where $A_{a_k} \in \mathcal{F}_{a_k}$ and define a function

$$
\begin{equation}
\begin{split}
\mu_{\Omega}^{\#} : \Omega &\to *\mathbb{R}_{c}^{\#} \cup \{ *\infty \} := \\
&\quad \mu^{\#}(A_{a_1}) \times \mu^{\#}(A_{a_2}) \times \dots \times \mu^{\#}(A_{a_n}) \times \left[ \mathrm{Ext-} \prod_{\substack{a \in \Delta, a \neq a_k, 1 \le k \le n \\ a \neq a}} \mu^{\#}(X_a) \right].
\end{split}
\tag{5.3.2}
\end{equation}
$$

This function possesses an extension (by #-additivity) on the #-algebra A generated
by Ω. It is easily to show that μΩ# is a #-premeasure on A.

**Definition 5.3.2** The #-measure $\hat{\mu}^\#$ on the $\sigma^\#$-algebra $\Sigma$ generated by $A$ accordingly
to **Theorem 2.1.3** is called the product #-measure of $\{\mu_{\alpha f}^\#\}_{\alpha \in \Delta}$, and the triple

$$
\left( \prod_{a \in \Delta} X_a, \Sigma, \hat{\mu}^{\#} \right) \text{ is called the product of #-measure spaces } (X_a, \Sigma_a, \mu_a^{\#}).
$$

We denote the $\sigma^\#$-algebra $\Sigma$ by $\bigotimes_{\alpha \in \Delta} \Sigma_\alpha$, and the #-measure $\hat{\mu}^\#$ by $\bigotimes_{\alpha \in \Delta} \mu_\alpha^\#$.

**Definition 5.3.3.**If $E \subseteq X_1 \times X_2$ and $x_1 \in X_1, x_2 \in X_2$, we define

$$
E_{x_1} = \{x \in X_2 : (x_1, x) \in E\} \text{ and } E^{x_2} = \{x \in X_1 : (x, x_2) \in E\}.
$$

If $f: X_1 \times X_2 \to *\mathbb{R}_c^\#$ is a function, we define $f_{x_1}: X_2 \to *\mathbb{R}_c^\#$ and $f^{x_2}: X_1 \to *\mathbb{R}_c^\#$ by $f_{x_1}(x) = f(x_1, x)$ and $f^{x_2}(x) = f(x, x_2)$.

**Theorem 5.3.1.** (The generalized Fubini's theorem) Let $\mu_1^\#$, $\mu_2^\#$ be $\sigma^\#$-hyperfinite
#-measures on $(X_1, \mathcal{F}_1)$ and $(X_2, \mathcal{F}_2)$,

$$ (X_1 \times X_2, \mathcal{F}_1 \otimes \mathcal{F}_2, \mu_1^{\#} \otimes \mu_2^{\#}) = (X_1, \mathcal{F}_1, \mu_1^{\#}) \times (X_2, \mathcal{F}_2, \mu_2^{\#}), \qquad (5.3.3) $$

and let $f \in L_1^{\#}(X_1 \times X_2, \mathcal{F}_1 \otimes \mathcal{F}_2, \mu_1^{\#} \otimes \mu_2^{\#})$. Then $f_{x_1} \in L_1^{\#}(X_2, \mathcal{F}_2, \mu_2^{\#})$, $\mu_1^{\#}$-#-a.e., and $f_{x_2} \in L_1^{\#}(X_1, \mathcal{F}_1, \mu_1^{\#})$, $\mu_2^{\#}$-#-a.e., and

$$ \begin{aligned} \mathrm{Ext}-\int_{X_1 \times X_2} f d^{\#}(\mu_1^{\#} \otimes \mu_2^{\#}) &= \mathrm{Ext}-\int_{X_2} \left[ \mathrm{Ext}-\int_{X_1} f^{x_2} d^{\#} \mu_1^{\#} \right] d^{\#} \mu_2^{\#} = \\ &= \mathrm{Ext}-\int_{X_1} \left[ \mathrm{Ext}-\int_{X_2} f_{x_1} d^{\#} \mu_2^{\#} \right] \end{aligned} \qquad (5.3.4) $$

**Lemma 5.3.1.** Let $(X_1, \Sigma_1, \mu_1^{\#})$ and $(X_2, \Sigma_2, \mu_2^{\#})$ be #-measure spaces, $E \in \Sigma_1 \otimes \Sigma_2$, and let $f$ be a $\Sigma_1 \otimes \Sigma_2$-measurable function on $X_1 \times X_2$, then:

(a) $E_{x_1} \in \Sigma_2$ for all $x_1 \in X_1$ and $E_{x_2} \in \Sigma_1$ for all $x_2 \in X_2$;

(b) $f_{x_1}$ is $\Sigma_2$-measurable and $f_{x_2}$ is $\Sigma_1$-measurable for all $x_1 \in X_1$ and $x_2 \in X_2$.

**Proof.** Denote by $A$ the collection of all $A \subseteq X_1 \times X_2$ such that $A_{x_1} \in \Sigma_2$ and $A^{x_2} \in \Sigma_1(\forall x_1 \in X_1, x_2 \in X_2)$.

The family $A$ contains all rectangles. Thus, since

$$ \left[ \bigcup_{n=1}^{\infty} A_n \right]_{x_1} = \bigcup_{n=1}^{\infty} [A_n]_{x_1}, [B_n]^{x_2} = [B_n]^{x_2} \qquad (5.3.5) $$

and

$$ [X_1 \times X_2 \setminus A]_{x_1} = X_2 \setminus A_{x_1}, [X_1 \times X_2 \setminus A]^{x_2} = X_1 \setminus A^{x_2}, \qquad (5.3.6) $$

$A$ is a $\sigma^{\#}$-algebra. So $\Sigma_1 \otimes \Sigma_2 \subseteq A$, and (a) is proved. Now the part (b) follows from (a) due to $f_{x_1}^{-1}(A) = [f^{-1}(A)]_{x_1}$ and $[f^{x_2}]^{-1}(A) = [f^{-1}(A)]^{x_2}(\forall A \subseteq \mathbb{R}_c^{\#})$.

**Definition 5.3.4** A family $M \subseteq P(X)$ is called a monotone class if $M$ is closed under $*$-countable increasing unions and $*$-countable decreasing intersections.

**Lemma 5.3.2.** If $A \subseteq P(X)$ is an algebra then the monotone class generated by $A$ coincides with the $\sigma^{\#}$-algebra generated by $A$.

**Lemma 5.3.3.** Let $(X_1, \Sigma_1, \mu_1^{\#})$ and $(X_2, \Sigma_2, \mu_2^{\#})$ be #-measure spaces, $E \in \Sigma_1 \otimes \Sigma_2$. Then the functions $x_1 \to \mu_2^{\#}(E_{x_1})$ and $x_2 \to \mu_1^{\#}(E^{x_2})$ are #-measurable on $(X_1, \Sigma_1)$ and $(X_2, \Sigma_2)$, and

$$ \mu_1^{\#} \otimes \mu_2^{\#}(E) = \mathrm{Ext}-\int_{X_2} \mu_1^{\#}(E^{x_2}) d^{\#} \mu_2^{\#} = \mathrm{Ext}-\int_{X_1} \mu_2^{\#}(E_{x_1}) d^{\#} \mu_1^{\#}. \qquad (5.3.7) $$

**Proof.** First we consider the case when $\mu_1^{\#}$ and $\mu_2^{\#}$ are finite. Let $A$ be the family of all $E \in \Sigma_1 \otimes \Sigma_2$ for which (5.3.7) is true. If $E = A \times B$, then $\mu_1^{\#}(E^{x_2}) = \mu_1^{\#}(A)\chi_B(x_2)$ and $\mu_2^{\#}(E_{x_1}) = \mu_2^{\#}(B)\chi_A(x_1)$, so $E \in A$. By additivity, it follows that hyperfinite disjoint unions of rectangles are in $A$ so, by Lemma 5.3.2, bit will suffice to show that $A$ is a monotone class. If $(E_n)_{n=1}^{\infty}$ is an increasing hyper infinite sequence in $A$ and $E = \bigcup_{n=1}^{\infty} E_n$, then the function $f_n(x_2) = \mu_1^{\#}((E_n)^{x_2})$

are #-measurable and increase pointwise to $f(y) = \mu_1^{\#}(E^{x_2})$. Hence $f$ is #-measurable and, by the monotone convergence theorem,
$$ \begin{aligned} \mathrm{Ext} - \int_{X_2} \mu_1^{\#}(E^{x_2}) d\mu_2^{\#} &= \#\mathrm{lim}_{n \to *\infty} \mathrm{Ext} X_1 \int_{X_2} \mu_1^{\#}((E_n)^{x_2}) d\mu_2^{\#} = \\ & \#\mathrm{lim}_{n \to *\infty} \mu_1^{\#} \times \mu_2^{\#}(E_n) = \mu_1^{\#} \times \mu_2^{\#}(E). \end{aligned} \quad (5.3.8) $$

Likewise $\mu_1^\# \times \mu_2^\#(E) = \mathrm{Ext} - \int_{X_1} \mu_2^\#(E_x)d\mu_1^\#$, so $E \in A$. Similarly, if $(E_n)_{n=1}^* \text{ is a decreasing}$
hyper infinite sequence in $A$ and $E = \bigcap_{n=1}^* E_n$, the function $x_2 \to \mu_1^\#((E_1)^{x_2})$ is in
$L_1^\#(\mu_2^\#)$ because $\mu_1^\#((E_1)x_2) \le \mu_1^\#(X_1) < *\infty$ and $\mu_2^\#(X_2) < *\infty$, so the dominated
convergence theorem can be applied to show that $E \in A$. Thus, $A$ is a monotone
class, and the proof is complete for the case of finite #-measure spaces.
Finally, if $\mu_1^\#$ and $\mu_2^\#$ are $\sigma^\#$-finite, we can write $X_1 \times X_2$ as the union of an
increasing hyper infinite sequence $(X_1^j \times X_2^j)_{j=1}^* \text{ of rectangles of finite or hyperfinite}$
#-measure. If $E \in \Sigma_1 \otimes \Sigma_2$, the preceding argument applies to $E \cap (X_1^j \times X_2^j)$ for each
$j$ gives us

$$ \mu_1^\# \times \mu_2^\#(E \cap (X_1^j \times X_2^j)) = \mathrm{Ext} - \int_{X_2} \mu_1^\#(E^{x_2} \cap X_1^j) \mu_2^\# = \mathrm{Ext} - \int_{X_1} \mu_2^\#(E_{x_1} \cap X_2^j) \mu_1^\#. \quad (5.3.9) $$

The application of the monotone convergence theorem then yields the desired
result.

**Lemma 5.3.3.** (Generalized Tonelli's theorem) Let $(X_1, \Sigma_1, \mu_1^\#)$ and $(X_2, \Sigma_2, \mu_2^\#)$ be #-measure spaces, and $f: X_1 \times X_2 \to *\mathbb{R}_{c+}^*$ be a $\Sigma_1 \otimes \Sigma_2$-#-measurable function. Then the functions

$$ f_{\mu_2^\#}(x_1) = \mathrm{Ext} - \int_{X_2} f_{x_1} d^\# \mu_2^\# \quad \text{and} \quad f_{\mu_1^\#}(x_2) = \mathrm{Ext} - \int_{X_1} f^{x_2} d^\# \mu_1^\# \quad (5.3.10) $$

are $\Sigma_1$-#-measurable and $\Sigma_2$-#-measurable, respectively, and

$$ \begin{aligned} \mathrm{Ext} - \int_{X_1 \times X_2} f d^\# \mu_1^\# \otimes \mu_2^\# &= \mathrm{Ext} - \int_{X_2} \left[ \mathrm{Ext} - \int_{X_1} f^{x_2} d^\# \mu_1^\# \right] d^\# \mu_2^\# = \\ &= \mathrm{Ext} - \int_{X_1} \left[ \mathrm{Ext} - \int_{X_2} f_{x_1} d^\# \mu_2^\# \right] d^\# \mu_1^\#. \end{aligned} \quad (5.3.11) $$

**Proof:** In the case when $f$ is a characteristic function, the statement of this lemma follows from Lemma 5.3.3. Therefore, by linearity, it holds also for nonnegative simple functions. If a nonnegative #-measurable function $f$ is arbitrary, there exists a sequence of nonnegative simple functions which increase pointwise to $f$, say $(f_n)_{n=1}^*$. By the monotone convergence theorem,

$$ \begin{aligned} \mathrm{Ext} - \int_{X_1} f_{\mu_2^\#} d^\# \mu_1^\# &= \#\mathrm{lim}_{n \to *\infty} \left[ \mathrm{Ext} - \int_{X_1} f_{\mu_2^\#}^n d^\# \mu_1^\# \right] = \\ &= \#\mathrm{lim}_{n \to *\infty} \left[ \mathrm{Ext} - \int_{X_1 \times X_2} f_n d^\# \mu_1^\# \otimes \mu_2^\# \right] \end{aligned} \quad (5.3.12) $$
and

$$
\begin{align}
\mathrm{Ext}-\int_{X_2} f_{\mu_1^{\#}} d^{\#} \mu_2^{\#} &= \#\lim_{n \to *\infty} \left[ \mathrm{Ext}-\int_{X_2} f_{\mu_1^{\#}}^n d^{\#} \mu_2^{\#} \right] = \tag{5.3.13} \\
&= \#\lim_{n \to *\infty} \left[ \mathrm{Ext}-\int_{X_1 \times X_2} f_n d^{\#} \mu_1^{\#} \otimes \mu_2^{\#} \right], \nonumber
\end{align}
$$

where

$$
f_{\mu_2^{\#}}^n(x_1) = \mathrm{Ext}-\int_{X_1} [f_n]_{x_1} d^{\#} \mu_2^{\#}, f_{\mu_1^{\#}}^n(x_2) = \mathrm{Ext}-\int_{X_2} [f_n]^{x_2} d^{\#} \mu_1^{\#}. \quad (5.3.14)
$$

This proves (5.3.11) and the lemma.

**Proof** of Theorem 5.3.1. Since an $*\mathbb{R}_c^{\#n}$-valued function $f$ is Lebesgue #-integrable iff its positive $f^+$ and negative $f^-$ parts are #-integrable, it is sufficient to prove the theorem only for nonnegative function $f \in L_1^\#(X_1 \times X_2, \Sigma, \mu_1^\# \otimes \mu_2^\#)$.

But this was exactly done in Lemma 5.3.3.

§ 5.4. Lebesgue #-measure and integral in *R<sup>#n</sup>*.

In this section, we study *R<sup>#n</sup>*<sub>c</sub>, n ∈ *N and functions from *R<sup>#n</sup>*<sub>c</sub> to *R<sup>#</sup>*<sub>c</sub> from the
point of view of the Lebesgue #-measure and Lebesgue integration. All results
presented below possess obvious *C<sup>#n</sup>*<sub>c</sub>-valued analogs. Then we define and study
generalized Cantor sets which are interesting from the point of view of the set
topology and the #-measure theory. Cantor sets are #-closed #-Borel nowhere
#-dense subsets of the interval [0, 1] or, more generally, of a Hausdorff #-space.

**Definition 5.4.1.** The Lebesgue #-measure $\mu^{\#n}$ on $*\mathbb{R}_c^{\#n}$ is the #-completion of the product of the Lebesgue #-measure on $*\mathbb{R}_c^{\#n}$ according to Definition 5.3.1. The domain $\Sigma^n$ of $\mu^\#$ (of course, $B^\#(*\mathbb{R}_c^{\#n}) \subseteq \Sigma^n$) is the class of Lebesgue #-measurable sets in $*\mathbb{R}_c^{\#n}$. We write $d^\# x^n$ for $d^\# \mu^{\#n}$ and

$$
\mathrm{Ext}-\int f(x)d^{\#}x^n = \mathrm{Ext}-\int fd^{\#}\mu^{\#n}.
$$

We extend some of the results of previous section to the *n*-dimensional case with

$n \in *\mathbb{N}$. If $E = \mathrm{Ext}-\prod_{j=1}^n E_j$ is a block in $*\mathbb{R}_c^{\#n}$, we call sets $E_j \subseteq *\mathbb{R}_c^{\#n}$ the sides of

the block E.

Theorem 5.4.1. Let $E \in \Sigma^n$. Then

(a) $\mu^{\#n}(E) = \inf\{\mu^{\#n}(U) : E \subseteq U, U \text{ #-open}\} = \sup\{\mu^{\#n}(K) : K \subseteq E, K \text{ #-compact}\}$;

(b) $E = A_1 \cup N_1 = A_2 \cup W_2$, where $A_1$ is an $F_{\sigma^#}$ set, $A_2$ is a $G_{\delta^#}$ set, and
$\mu^{\#n}(N_1) = \mu^{\#n}(N_2) = 0$;

(c) If $\mu^{\#n}(E) < *\infty$ then, for any $\varepsilon \approx 0, \varepsilon > 0$, there is a hyperfinite family $\{R_j\}_{j=1}^N$ of disjoint blocks, whose sides are intervals such that $\mu^{\#n}(E\Delta \cup_{j=1}^N R_j) < \varepsilon$.

**Proof:** By the definition of product #-measures, if $E \in \Sigma^n$ and $\varepsilon \approx 0, \varepsilon > 0$, there is a *-countable family* $\{T_j\}_{j=1}^{*\infty}$ of blocks such that $E \subseteq \cup_{j=1}^{*\infty} T_j$ and

$$
\mathrm{Ext}-\sum_{j=1}^{*\infty} \mu^{\#n}(T_j) \leq \mu^{\#n}(E) + \varepsilon.
$$

For each *j*, by applying Theorem 5.2.3 to the sides of *R*<sub>*j*</sub>, we can find blocks
$U_j \supseteq F_j$ whose sides are #-open sets such that $\mu^{\#n}(U_j) < \mu(T_j) + \varepsilon 2^{-j}$.

If $U = \bigcup_{j=1}^{\infty} U_j$ then $U$ is #-open and

$$ \mu^{\#n}(U) \leq \mathrm{Ext} - \sum_{j=1}^{\infty} \mu^{\#n}(U_j) \leq \mu^{\#n}(E) + 2\varepsilon. $$

This proves the first equation in part (a). The second equation and part (b) follow as in the proofs of Theorems 2.1.6 and 2.1.7.

Next, if $\mu^{\#n}(E) < \infty$ then $\mu^{\#n}(U_j) < \infty$ for all $j$. Since the sides of $U_j$ are $*$-countable unions of #-open intervals, by taking suitable hyperfinite subunions, we obtain blocks $V_j \subseteq U_j$ whose sides are hyperfinite unions of intervals such that $\mu^{\#n}(V_j) \geq \mu^{\#n}(U_j) - \varepsilon 2^{-j}$. If $N \in \mathbb{N}$ is sufficiently hyperfinite large, we have

$$ \mu^{\#n} \left( E \setminus \bigcup_{j=1}^{N} V_j \right) \leq \mu^{\#n} \left( \bigcup_{j=1}^{N} U_j \setminus V_j \right) + \mu^{\#n} \left( \bigcup_{j=N+1}^{\infty} U_j \right) < 2\varepsilon $$

and

$$ \mu^{\#n} \left( \bigcup_{j=1}^{N} V_j \setminus E \right) \leq \mu^{\#n} \left( \bigcup_{j=1}^{\infty} U_j \setminus E \right) < \varepsilon, $$

so $\mu^{\#n}(E\Delta \bigcup_{j=1}^{N} V_j) < 3\varepsilon$. Since $\bigcup_{j=1}^{N} V_j$ can be expressed as a hyperfinite disjoint union of rectangles whose sides are intervals, we have proved (c).

## § 5.5. Lebesgue #-integrable functions on $\mathbb{R}^{\#n}$

Let $\mu^{\#n}$ be the Lebesgue #-measure in $\mathbb{R}^{\#n}$. The set $M(*\mathbb{R}_c^{\#n}, \mu^{\#n})$ of all $*\mathbb{R}_c^{\#n}$-valued $\mu^{\#n}$-measurable functions on $*\mathbb{R}_c^{\#n}$ is a vector space (addition and scalar multiplication are pointwise). By $L_1^1(*\mathbb{R}_c^{\#n}, \mu^{\#n})$ we denote its subspace of all Lebesgue #-integrable functions (with finite in $*\mathbb{R}_c^{\#n}$-integral). Now write $f \approx g$ for $f$ and $g$ in $M(*\mathbb{R}_c^{\#n}, \mu^{\#n})$, whenever $f$ and $g$ differ only on a $\mu^{\#n}$-null set (a set of $\mu^{\#n}$-measure zero). It is easily seen that $\approx$ is an equivalencerelation. Let $L_0 = L_0(*\mathbb{R}_c^{\#n}, \mu^{\#n})$ be the set of equivalence classes of functions in $M(*\mathbb{R}_c^{\#n}, \mu^{\#n})$. We denote the equivalence classes of $f, g, \dots$ by $[f], [g], \dots$. The set $L_0$ becomes a vector space over field $*\mathbb{R}_c^\#$ by defining $[f] + [g] = [f + g]$ and $\alpha[f] = [\alpha f]$ for a real $\alpha \in *\mathbb{R}_c^\#$. Observe that these definitions do not depend on the choice of $f$ and $g$ in their equivalence classes. The same is true for the partial order in $L_0$, if we define $[f] \leq [g]$ to mean $f(x) \leq g(x)$ for all $x \in *\mathbb{R}_c^{\#n}$ except a null set. In practice, the elements of $L_0 = L_0(*\mathbb{R}_c^{\#n}, \mu^{\#n})$ are usually denoted by $f, g, \dots$ and treated as if they were functions instead of equivalence classes of functions.

### Definition 5.5.1.

**Theorem 5.5.1.** If $f \in L_1^1(\mu^{\#n})$ and $\varepsilon \approx 0, \varepsilon > 0$, there is a simple function $\varphi = \mathrm{Ext} - \sum_{j=1}^N \alpha_j \chi_{R_j}$, where each $R_j$ is a product of intervals such that $\mathrm{Ext} - \int |f - \varphi| d^\# \mu^{\#n} < \varepsilon$, and there is a #-continuous function $g$ vanishing outside of a bounded in $*\mathbb{R}_c^{\#n}$ set such that $\mathrm{Ext} - \int |f - g| d^\# \mu^{\#n} < \varepsilon < \varepsilon$.

**Proof.** By the definition of Lebesgue #-integrable functions, we can approximate $f$ by simple functions in $L_1^1$-norm. Then use Theorem 5.4.1 to approximate a simple function by a function $\varphi$ of the desired form. Finally, use the generalized Urysohn Lemma to approximate such $\varphi$ by a #-continuous function.

**Theorem 5.5.2.** The Lebesgue #-measure on $*\mathbb{R}_c^{\#n}$ is translation-invariant. Namely, let $a \in *\mathbb{R}_c^{\#n}$. Define the shift $\tau_a : *\mathbb{R}_c^{\#n} \to *\mathbb{R}_c^{\#n}$ by $\tau_a(x) = x + a$.

(a) If $E \in \mathcal{L}^{\#n}$ then $\tau_a(E) \in \mathcal{L}^{\#n}$ and $\mu^{\#n}(\tau_a(E)) = \mu^{\#n}(E)$;
(b) If $f : *\mathbb{R}_c^{\#n} \to *\mathbb{R}_c^\#$ is Lebesgue #-measurable then so is $f \circ \tau_a$. Moreover, if either $f \ge 0$ or $f \in L_1^\#(\mu^{\#n})$ then

$$ \mathrm{Ext} - \int (f \circ \tau_a) d^\# \mu^{\#n} = \mathrm{Ext} - \int f d^\# \mu^{\#n}. \qquad (5.5.1) $$

**Proof.** Since $\tau_a$ and its inverse $\tau_{-a}$ are #-continuous, they preserve the class of #-Borel sets. The formula $\mu^{\#n}(\tau_a(E)) = \mu^{\#n}(E)$ follows easily from the trivial one dimensional variant of this result if $E$ is a block. For a general #-Borel set $E$, the formula $\mu^{\#n}(\tau_a(E)) = \mu^{\#n}(E)$ follows from the previous step, since $\mu^{\#n}$ is determined by its action on blocks. Assertion (a) now follows immediately.

If $f$ is Lebesgue #-measurable and $B$ is a #-Borel set in $*\mathbb{R}_c^\#$, we have $f^{-1}(B) = E \cup N$, where $E$ is #-Borel and $\mu^\#(N) = 0$. But $\tau_a^{-1}(E)$ is Borel and $\mu^\#(\tau_a^{-1}(N)) = 0$, so $(f \circ \tau_a)^{-1}(B) \in \Sigma^n$ and $f$ is Lebesgue #-measurable. The equality (5.5.1) reduces to the equality $\mu^{\#n}(\tau_{-a}(E)) = \mu^{\#n}(E)$ when $f = \chi_E$. It is true for simple functions by linearity, and hence for nonnegative #-measurable functions by the definition of #-integral. Taking positive and negative parts of real and imaginary parts, we obtain the result for $f \in L_1^\#(\mu^{\#n})$.

**Theorem 5.5.3.** (Generalized Lusin's theorem) If $f$ is a Lebesgue #-measurable function on $*\mathbb{R}_c^\#$ and $\varepsilon \approx 0, \varepsilon > 0$ then there exist a #-measurable set $A \subseteq *\mathbb{R}_c^\#$ such that $\mu^{\#n}(*\mathbb{R}_c^\# \setminus A) \le \varepsilon$ and the restriction of $f$ onto $A$ is #-continuous.

# Chapter II. $*\mathbb{R}_c^\#$-valued distributions.

## §1. $*\mathbb{R}_c^\#$-valued test functions and distributions

Definitions and theorems appropriate to analysis on non-Archemedean field $*\mathbb{R}_c^\#$ and on complex field $*\mathbb{C}_c^\# = *\mathbb{R}_c^\# + i*\mathbb{R}_c^\#$ are given in [1]-[2].

**Definition 1.1.** [3].(i) Let $U$ be a free ultrafilters on $\mathbb{N}$ and introduce an equivalence relation on sequences in $\mathbb{R}^\mathbb{N}$ as $f_1 \sim_U f_2$ iff $\{i \in \mathbb{N} | f_1(i) = f_2(i)\} \in U$.

(ii) $\mathbb{R}^\mathbb{N}$ divided out by the equivalence relation $\sim_U$ gives us the nonstandard extension $*\mathbb{R}$, the hyperreals; in symbols, $*\mathbb{R} = \mathbb{R}^\mathbb{N}/\sim_U$ and similarly $\mathbb{N}^\mathbb{N}$ divided out by the equivalence relation $\sim_U$ gives us the nonstandard extension $*\mathbb{N}$, the hyperintegers; in symbols, $*\mathbb{N} = \mathbb{N}^\mathbb{N}/\sim_U$.

**Abbreviation 1.1.** If $f \in \mathbb{R}^\mathbb{N}$, we denote its image in $*\mathbb{R}$ by $[f]$, i.e., $[f] = \{g \in \mathbb{R}^\mathbb{N} | g \sim_U f\}$.

**Remark 1.1.** For any real number $r \in \mathbb{R}$ let $\mathbf{r}$ denote the constant function $\mathbf{r} : \mathbb{N} \to \mathbb{R}$ with value $r$, i.e., $\mathbf{r}(n) = r$, for all $n \in \mathbb{N}$. We then have a natural embedding

$$ *\langle \cdot \rangle : \mathbb{R} \hookrightarrow *\mathbb{R} $$

by setting $*r = [\mathbf{r}(n)]$ for all $r \in \mathbb{R}$. We denote its image $*(\mathbb{R})$ in $*\mathbb{R}$ by $*\mathbb{R}_{\mathrm{st}}$.

**Definition 1.2.** [3]. An element $x \in *\mathbb{R}$ is called finite if $|x| < r$ for some $r \in \mathbb{Q}, r > 0$.

**Abbreviation 1.2.** For $x \in *\mathbb{R}$ we abbreviate $x \in *\mathbb{R}_{\mathrm{fin}}$ if $x$ is finite.

**Remark 1.2.** [3]. Let $x \in *\mathbb{R}_{\mathrm{fin}}$ be finite. Let $D_1$, be the set of $r \in \mathbb{Q}$ such that $r < x$ and $D_2$ the set of $r' \in \mathbb{Q}$ such that $x < r'$. The pair $(D_1, D_2)$ forms a Dedekind cut in $\mathbb{R}$, hence determines a unique $r_0 \in \mathbb{R}$. A simple argument shows that $|x - r_0|$ is infinitesimal, i.e., $|x - r_0| \approx 0$.

**Definition 1.3.** [1]. This unique $r_0$ is called the standard part of $x$ and is denoted by $\circ x$ or $st(x)$.

The following notation will be used throughout this paper.

$n \in \mathbb{N}^\#$ is a fixed positive integer and $U \subset *\mathbb{R}_c^{\#n}$ is a fixed non-empty #-open subset of
Inear space $\mathbb{R}_c^{\#n}$ over non Archemedan field$^*$ $\mathbb{R}_c^\#$.

$\mathbb{N} = \{0, 1, 2, \dots\}$ denotes the standard natural numbers.

$k$ will denote a non-negative integer or $\infty^\#$.

If $f$ is a function then $\mathbf{Dom}(f)$ will denote its domain and the support of $f$, denoted by $\text{supp}(f)$, is defined to be the closure of the set $\{x \in \mathbf{Dom}(f) : f(x) \neq 0\}$ in $\mathbf{Dom}(f)$.
For two functions $f,g : U \to \mathbb{C}_c^\#$, the following notation defines external canonical pairing:

$$ \langle f, g \rangle = \mathrm{Ext} - \int_U f(x)g(x)d^\#x. \qquad (1.1) $$

A multi-index of size $n \in \mathbb{N}^\#$ is an element in $\mathbb{N}^{\#n}$, if the size of multi-indices is omitted then the size should be assumed to be $n$. The length of a multi-index $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^{\#n}$ is defined as $\mathrm{Ext}-\sum_{i=1}^n \alpha_i$ and denoted by $|\alpha|$. Multi-indices are particularly useful when dealing with functions of several variables, in particular we introduce the following canonical notations for a given multi-index $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^{\#n}$,

$$ \begin{gathered} x^{\alpha} = x_1^{\alpha_1} \cdots x_n^{\alpha_n}, \\ \partial^{\#\alpha} = \frac{\partial^{\#|\alpha|}}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} \end{gathered} \qquad (1.2) $$

We also introduce a partial order of all multi-indices by $\beta \ge \alpha$ if and only if $\beta_i \ge \alpha_i$ for all $1 \le i \le n$. When $\beta \ge \alpha$ we define their multi-index binomial coefficient as:

$$ \binom{\beta}{\alpha} = \binom{\beta_1}{\alpha_1} \cdots \binom{\beta_n}{\alpha_n}. $$

1. Let $k \in \mathbb{N}^\# \cup \infty^\#$.

2. Let $C^{\#k}(U)$ denote the vector space of all $k$-times $\#$-continuously $\#$-differentiable $^*\mathbb{R}_c^\#$-valued or $^*\mathbb{C}_c^\#$-valued functions on $U$.

For any $\#$-compact subset $K \subseteq U$, let $C^{\#k}(K)$ and $C^{\#k}(K; U)$ both denote the vector space of all those functions $f \in C^{\#k}(U)$ such that $\mathrm{supp}(f) \subseteq K$.

Note that $C^{\#k}(K)$ depends on both $K$ and $U$ but we will only indicate $K$, where in particular, if $f \in C^{\#k}(K)$ then the domain of $f$ is $U$ rather than $K$. We will use the notation

$C^{\#k}(K; U)$ only when the notation $C^{\#k}(K)$ risks being ambiguous.

Every $C^{\#k}(K)$ contains the constant 0 map, even if $K = \emptyset$.

Let $C_c^{\#k}(U)$ denote the set of all $f \in C^{\#k}(U)$ such that $f \in C^{\#k}(K)$ for some $\#$-compact subset $K$ of $U$.

Equivalently, $C_c^{\#k}(U)$ is the set of all $f \in C^{\#k}(U)$ such that $f$ has $\#$-compact support.

$C_c^{\#k}(U)$ is equal to the union of all $C^{\#k}(K)$ as $K \subseteq U$ ranges over all $\#$-compact subsets of $U$. If $f$ is a $^*\mathbb{R}_c^\#$-valued function on $U$, then $f$ is an element of $C_c^{\#k}(U)$ if and only if $f$ is a $C^{\#k}$ bump function. Every $^*\mathbb{R}_c^\#$-valued test function on $U$ is always also a $^*\mathbb{C}_c^\#$-valued test function on $U$.

For all $j,k \in \mathbb{N}$ and any $\#$-compact subsets $K$ and $L$ of $U$, we have:

$$ C^{\#k}(K) \subseteq C_c^{\#k}(U) \subseteq C^{\#k}(U); $$

$$ C^{\#k}(K) \subseteq C^{\#k}(L) \text{ if } K \subseteq LC^{\#k}(K) \subseteq C^{\#j}(K) \text{ if } j \le k; $$

$$ C_c^{\#k}(U) \subseteq C_c^{\#j}(U) \text{ if } j \le k; $$

$$ C^{\#k}(U) \subseteq C^{\#j}(U) \text{ if } j \le k. $$
**Definition1.1.** Elements of $C_c^{\#\infty}(U)$ are called $*\mathbb{R}_c^\#$-valued test functions on $U$ and $C_c^{\#\infty}(U)$ is

called the space of $*\mathbb{R}_c^\#$-valued test functions on $U$. We will use both $D(U)$ and $C_c^{\#\infty}(U)$ to denote this space.

**Definition1.2.** Distributions on $U$ are #-continuous $*\mathbb{R}_c^\#$-valued linear functionals on $C_c^{\#\infty}(U)$ when this vector space is endowed with a particular topology called the canonical

LF-topology.

The following proposition states two necessary and sufficient conditions for the #-continuity of a linear functional on $C_c^{\#\infty}(U)$ that are often straightforward to verify.

**Proposition1.1.** A linear functional $T$ on $C_c^{\#\infty}(U)$ is #-continuous, and therefore a distribution, if and only if either of the following equivalent conditions are satisfied:

1. For every #-compact subset $K \subseteq U$ there exist constants $C > 0$ and $N \in \mathbb{N}$ dependent on $K$ such that for all $f \in C_c^{\#\infty}(U)$ with support contained in $K$ $|T(f)| \le C \sup\{|\partial^{\#a}f(x)|: x \in U, |\alpha| \le N\}$.

2. For every #-compact subset $K \subseteq U$ and every sequence $\{f_i\}_{i=1}^\infty$ in $C_c^{\#\infty}(U)$ whose supports are contained in $K$, if $\{\partial^{\#a}f_i\}_{i=1}^\infty$ #-converges uniformly to zero on $U$ for every multi-index $\alpha$, then $\#\lim_{i \to \infty} T(f_i) = 0$.

# 2. The non-Archimedean external $*\mathbb{R}_c^\#$-Valued Schwartz distributions.

Defined below are the tempered distributions, which form a subspace of $D^{\#'}(*\mathbb{R}_c^{\#n})$, the space of distributions on $*\mathbb{R}_c^{\#n}$. This is a proper subspace: while every tempered distribution is a distribution and an element of $D^{\#'}(*\mathbb{R}_c^{\#n})$ the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in $D^{\#'}(*\mathbb{R}_c^{\#n})$.

## § 2.1. Schwartz space $S^\#(*\mathbb{R}_c^{\#n})$.

**Definition 2.1.** A function $f: X \to *\mathbb{R}_c^\#$ defined on some set $X$ is called **finitely bounded** (or **bounded**) if the set of its values is finitely bounded, i.e., $f(X) \subset [a, b]$ where $a, b \in *\mathbb{R}_c^{\#\text{fin}}$. In other words, there exists a finite hyperreal number $M \in *\mathbb{R}_c^{\#\text{fin}}$ such that

$$|f(X)| \le M. \qquad (2.1)$$

**Definition 2.2.** A function $f: X \to *\mathbb{R}_c^\#$ defined on some set $X$ is called **hyper finitely bounded** (or **hyper bounded**) if the set of its values is hyper finitely bounded, i.e., $f(X) \subset [a, b]$ where $a, b \in *\mathbb{R}_c^{\#} \setminus *\mathbb{R}_c^{\#\text{fin}}$. In other words, there exists a hyperfinite hyperreal number $M \in *\mathbb{R}_c^{\#} \setminus *\mathbb{R}_c^{\#\text{fin}}$ such that $|f(X)| \le M$.

**Definition 2.3.** For $n \in \mathbb{N}^\#$, an #-integrable function $\phi: *\mathbb{R}_c^{\#n} \to *\mathbb{R}_c^\#$ is called #-rapidly decreasing if for all $\alpha \in \mathbb{N}^{\#n}$ the product function $x \mapsto x^\alpha \phi(x)$ is a finitely bounded or hyper finitely bounded function.

**Remark 2.1.** If $\phi$ is a #-rapidly decreasing function, then its integral exists
$$ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} \phi(x) d^{\#n}x < \infty^{\#} \qquad (2.2) $$

In fact for all $\alpha \in \mathbb{N}^{\#n}$ the integral of $x \mapsto x^{\alpha}\phi(x)$ exists

$$ \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} x^{\alpha}\phi(x) d^{\#n}x < \infty^{\#}. \qquad (2.3) $$

**Definition 2.4.** The Schwartz space, $S^{\#}(*\mathbb{R}_c^{\#n})$, is the space of all #-smooth functions in $C^{\#\infty}(*\mathbb{R}_c^{\#n})$ that are rapidly decreasing at #-infinity along with all partial #-derivatives. Thus

$\phi : *\mathbb{R}_c^{\#n} \to *\mathbb{R}_c^{\#n}$ is in the Schwartz space provided that any #-derivative of $\phi$, multiplied with any power of $|x|$, #-converges to 0 as $|x| \to \infty^{\#}$. These functions form a #-complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices $\alpha$ and $\beta$ define:

$$ p_{\alpha,\beta}(\phi) = \sup_{x \in *\mathbb{R}_c^{\#n}} |x^{\alpha} \hat{\partial}^{\#\beta} \phi(x)|. \qquad (2.1) $$

Then $\phi$ is in the Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$ if all the values satisfy: $p_{\alpha,\beta}(\phi) < \infty^{\#}$. Thus

$$ S^{\#}(*\mathbb{R}_c^{\#n}, *\mathbb{R}_c^{\#n}) \triangleq \{\phi \in C^{\infty}(*\mathbb{R}_c^{\#n}, *\mathbb{R}_c^{\#n}) | \forall \alpha, \beta \in \mathbb{N}^{\#n} (p_{\alpha,\beta}(\phi) < \infty^{\#})\}. $$

Similarly

$$ S^{\#}(*\mathbb{R}_c^{\#n}, *\mathbb{C}_c^{\#n}) \triangleq \{\phi \in C^{\infty}(*\mathbb{R}_c^{\#n}, *\mathbb{C}_c^{\#n}) | \forall \alpha, \beta \in \mathbb{N}^{\#n} (p_{\alpha,\beta}(\phi) < \infty^{\#})\} $$

The family of seminorms $p_{\alpha,\beta}(\cdot)$ defines a locally convex topology on the Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$.

For $n=1$, the seminorms are norms on the Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$. One can also use the following family of seminorms to define the topology:

$$ |f|_{m,k} = \sup_{|p| \le m} \left( \sup_{x \in *\mathbb{R}_c^{\#n}} \{(1 + |x|)^k |(\hat{\partial}^{\#\alpha} f)(x)|\} \right), k, m \in \mathbb{N}^{\#}. \qquad (2.2) $$

Otherwise, one can define a norm on $S^{\#}(*\mathbb{R}_c^{\#n})$ by

$$ \|\phi\|_k = \max_{|\alpha|+|\beta| \le k} \sup_{x \in *\mathbb{R}_c^{\#n}} |x^{\alpha} \partial^{\beta} \phi(x)|, k \ge 1. \qquad (2.3) $$

The Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$ is a Fréchet space (that is, a #-complete metrizable locally convex space). Because the Fourier transform changes $\hat{\partial}^{\#\alpha}$ into multiplication by $x^{\alpha}$ and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.

**Definition 2.5.** A sequence $\{f_i\}_{i=1}^{\infty}$ #-converges to 0 in $S^{\#}(*\mathbb{R}_c^{\#n})$ if and only if the functions $(1+|x|)^k(\hat{\partial}^{\#p}f_i)(x)$ #-converge to 0 uniformly in the whole of $*\mathbb{R}_c^{\#n}$, which implies that such a sequence must converge to zero in $C^{\#\infty}(*\mathbb{R}_c^{\#n})$.

The subset of all #-analytic Schwartz functions is #-dense in $S^{\#}(*\mathbb{R}_c^{\#n})$

The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms $S^{\#}(*\mathbb{R}_c^{\#m}) \hat{\otimes} S^{\#}(*\mathbb{R}_c^{\#n}) \to S^{\#}(*\mathbb{R}_c^{\#m+n})$, where $\hat{\otimes}$ represents the #-completion of the injective tensor product

## § 2.2. Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$

**Definition 2.6.** For $n \in \mathbb{N}$, an $*\mathbb{R}_{c,\mathrm{fin}}^{\#n}$-valued and #-integrable function
$\phi : *\mathbb{R}_c^{\#n} \to *\mathbb{R}_{c,fin}^{\#n}$ is called #-rapidly decreasing if for all $\alpha \in \mathbb{N}^n$ the product function $x \mapsto x^\alpha \phi(x)$ is a finitely bounded function.

**Remark 2.2.** If $\phi$ is a #-rapidly decreasing $*\mathbb{R}_{c,fin}^{\#n}$-valued function, then its integral exists and finite, i.e.,

$$ \mathrm{Ext}- \int_{*\mathbb{R}_c^{\#n}} \phi(x) d^{\#n} x \in *\mathbb{R}_{c,fin}^{\#n}. \qquad (2.2) $$

In fact for all $\alpha \in \mathbb{N}^n$ the integral of $x \mapsto x^\alpha \phi(x)$ exists and finite, i.e.,

$$ \mathrm{Ext}- \int_{*\mathbb{R}_c^{\#n}} x^\alpha \phi(x) d^{\#n} x \in *\mathbb{R}_{c,fin}^{\#n}. \qquad (2.3) $$

It follows from () that for all $\alpha \in \mathbb{N}^n$ and for any $R \in *\mathbb{R}_c^{\#n} \setminus *\mathbb{R}_{c,fin}^{\#n}$

$$ \mathrm{Ext}- \int_{*\mathbb{R}_c^{\#n} \setminus B(R)} x^\alpha \phi(x) d^{\#n} x \approx 0 \qquad (2.3) $$

where $B(R) \triangleq \{x \in *\mathbb{R}_c^{\#n} | x| \le R\}$

**Definition 2.7.** The Schwartz space, $S_{fin}^{\#}(*\mathbb{R}_c^{\#n})$, is the space of all $*\mathbb{R}_c^{\#n}$-valued #-smooth functions that are rapidly decreasing at #-infinity along with all partial #-derivatives any finite order $1 \le m < \infty$.

Thus

$\phi : *\mathbb{R}_c^{\#n} \to \mathbb{R}^\#$ is in the Schwartz space provided that any #-derivative of $\phi$, multiplied with any power of $|x|$, #-converges to 0 as $|x| \to \infty$. These functions form a #-complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices $\alpha$ and $\beta$ define:

$$ p_{\alpha,\beta}(\phi) = \sup_{x \in *\mathbb{R}_c^{\#n}} |x^\alpha \partial^{\#\beta} \phi(x)|. \qquad (2.1) $$

## § 2.3. Non-Archimedean tempered distributions $S^{\#}(*\mathbb{R}_c^{\#n})$.

A non-Archimedean tempered distribution is a distribution $u \in D'(*\mathbb{R}_c^{\#n})$ that does not "grow too fast" – at most polynomial (or tempered) growth – at #-infinity in all directions; in particular it is only defined on $*\mathbb{R}_c^{\#n}$, not on any #-open subset.

Formally, a tempered distribution is a #-continuous linear functional on the Schwartz space $S^{\#}(*\mathbb{R}_c^{\#n})$ of smooth functions with #-rapidly decreasing #-derivatives. The space of tempered distributions (with its natural topology) is denoted $S^{\#}(*\mathbb{R}_c^{\#n})$.

Every #-compactly supported distribution is a tempered distribution, yielding an inclusion $E^{\#}(*\mathbb{R}_c^{\#n}) \hookrightarrow S^{\#}(*\mathbb{R}_c^{\#n})$.

## § 3. The Fourier transform on $S^{\#}(*\mathbb{R}_c^{\#n})$, $S_{fin}^{\#}(*\mathbb{R}_c^{\#n})$

We begin by defining the Fourier transform, and the inverse transform, on $S^{\#}(*\mathbb{R}_c^{\#n})$,

$n \in \mathbb{N}^\#$, the Schwartz space of $C^\infty$ functions of rapid decrease.

**Definition 3.1.** Suppose $f \in S^{\#}(*\mathbb{R}_c^{\#n})$. The Fourier transform of $f(x)$ is the function $\hat{f}(\lambda)$ given by
$$ \hat{f}(\lambda) = \frac{1}{(2\pi\#)^{n/2}} \left( \mathrm{Ext} - \int_{\mathbb{R}_c^{\#n}} f(x)[\mathrm{Ext}\text{-exp}(-ix \cdot \lambda)]d^{\#n}x \right), \quad (3.1) $$

where $x \cdot \lambda = \mathrm{Ext}-\sum_{i=1}^n x_i \lambda_i$. The inverse Fourier transform of $f$, denoted by $\check{f}$, is the function

$$ \check{f}(\lambda) = \frac{1}{(2\pi\#)^{n/2}} \left( \mathrm{Ext} - \int_{\mathbb{R}_c^{\#n}} f(x)[\mathrm{Ext}\text{-exp}(ix \cdot \lambda)]d^{\#n}x \right). \quad (3.2) $$

We will usually write $\hat{f} = \mathcal{F}[f]$ and $\check{f} = \mathcal{F}^{-1}[f]$.

Since every function in Schwartz space is in $\mathcal{L}_1^\#(*\mathbb{R}_c^{\#n})$, the above integrals (1.1) and (1.2) make sense.

We will use the standard multi-index notation. A multi-index $\alpha = (\alpha_1, \dots, \alpha_n), n \in \mathbb{N}^\#$ is an $n$-tuple of nonnegative integers. The collection of all multi-indices will be denoted by $I_+^\alpha$. The symbols $|\alpha|, x^\alpha, D^{\#\alpha}$, and $x^2$ are defined as follows:

$$ |\alpha| = \mathrm{Ext}-\sum_{i=1}^n \alpha_i $$

$$ x^\alpha = \mathrm{Ext}-\prod_{i=1}^n x_i^{\alpha_i} \quad \text{or} \quad \mathrm{Ext}-(x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}) \quad \text{or} \quad \text{simbolically} \quad x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n} $$

$$ D^{\#\alpha} f(x) = \mathrm{Ext}-\prod_{i=1}^{n} \frac{\partial^{\#\alpha_i} f(x)}{\partial^{\#} x^{\alpha_i}} \quad \text{or simbolically} \quad D^{\#\alpha} f(x) = \frac{\partial^{\#|\alpha|} f(x)}{\partial^{\#} x^{\alpha_1} \partial^{\#} x^{\alpha_2} \cdots \partial^{\#} x^{\alpha_n}} \quad (3.3) $$

$$ x^2 = \mathrm{Ext}-\sum_{i=1}^{n} x_i^2. $$

**Lemma 1.1.** The maps $f \mapsto \hat{f}$ and $f \mapsto \check{f}$ are #-continuous linear transformations of $S^\#(*\mathbb{R}_c^{\#n})$ into $S^\#(*\mathbb{R}_c^{\#n})$. Furthermore, if $\alpha$ and $\beta$ are multi-indices, then

$$ ((i\lambda)^\alpha D^{\#\beta} \hat{f})(\lambda) = \overline{D^{\#\alpha} ((-ix)^\beta f(x))}(\lambda). \quad (3.4) $$

**Proof** The map $f \mapsto \hat{f}$ is clearly linear. Since

$$ ((i\lambda)^\alpha D^{\#\beta} \hat{f})(\lambda) = $$

$$ \frac{1}{(2\pi\#)^{n/2}} \left( \mathrm{Ext} - \int_{\mathbb{R}_c^{\#n}} (\lambda^\alpha)(-ix)^\beta f(x) [\mathrm{Ext}\text{-exp}(-ix\cdot\lambda)]f(x) d^{\#n}x \right) = $$

$$ \frac{1}{(2\pi\#)^{n/2}} \left( \mathrm{Ext} - \int_{\mathbb{R}_c^{\#n}} \frac{1}{(-i)^\alpha} (D_x^{\#\alpha} [\mathrm{Ext}\text{-exp}(-ix\cdot\lambda)]) (-ix)^\beta f(x) d^{\#n}x \right) = \quad (3.5) $$

$$ \frac{(-i)^\alpha}{(2\pi\#)^{n/2}} \left( \mathrm{Ext} - \int_{\mathbb{R}_c^{\#n} x}^{\#\beta} [\mathrm{Ext}\text{-exp}(-ix\cdot\lambda)] D_x^{\#\alpha} ((-ix)^\beta f(x)) d^{\#n}x \right). $$

We conclude that
$$ \|\hat{f}\|_{\alpha, \beta} = \sup_{\lambda \in *\mathbb{R}_c^{\#n}} |\lambda^\alpha (D^{\#\beta} \hat{f})(\lambda)| \le \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} |D_x^{\#\alpha}(x^\beta f(x))| d^{\#n}x \right) < \infty^\# \quad (3.6) $$

so $f \mapsto \hat{f}$ takes $S^#(*\mathbb{R}_c^{\#n})$ into $S^#(*\mathbb{R}_c^{\#n})$, and we have also proven (1.4). Furthermore, if $k$ is large enough, $\int (1+x^2)^{-k} d^{\#n}x < \infty^\#$ so that

$$ \begin{aligned} \|\hat{f}\|_{\alpha, \beta} &\le \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} \frac{(1+x^2)^{-k}}{(1+x^2)^{-k}} |D_x^{\#\alpha}((-ix)^\beta f(x))| d^{\#n}x \right) \le \\ &\frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} (1+x^2)^{-k} d^{\#n}x \right) \sup_{x \in *\mathbb{R}_c^{\#n}} \left\{ (1+x^2)^{+k} |D_x^{\#\alpha}((-ix)^\beta f(x))| \right\}. \end{aligned} \quad (3.7) $$

Using generalized Leibnitz's rule we easily conclude that there exist multi-indices $\alpha_j, \beta_j$ and constants $c_j$ so that

$$ \|\hat{f}\|_{\alpha, \beta} \le \sum_{j=1}^{M} c_j \|f\|_{\alpha_j, \beta_j}. \quad (1.8) $$

Thus the map $f \mapsto \hat{f}$ is bounded and therefore #-continuous. The proof for $f \mapsto \tilde{f}$ is the same.

**Theorem 1.1.** (Generalized Fourier inversion theorem) The Fourier transform (3.1) is a linear bicontinuous bijection from $S^#(*\mathbb{R}_c^{\#n})$ onto $S^#(*\mathbb{R}_c^{\#n})$. Its inverse map is the inverse Fourier transform, i.e., $\mathcal{F}^{-1}(\mathcal{F}[f]) = f$ and $\mathcal{F}(\mathcal{F}^{-1}[f]) = f$.

**Proof.** We will prove that $\mathcal{F}^{-1}(\mathcal{F}[f]) = f$. The proof that $\mathcal{F}(\mathcal{F}^{-1}[f]) = f$ is similar. $\mathcal{F}(\mathcal{F}^{-1}[f]) = f$ implies that $\mathcal{F}[f]$ is surjective and $\mathcal{F}^{-1}(\mathcal{F}[f]) = f$ implies that $\mathcal{F}[f]$ is injective. Since $\mathcal{F}[f]$ and $\mathcal{F}^{-1}[f]$ are #-continuous maps of $S^#(*\mathbb{R}_c^{\#n})$ onto $S^#(*\mathbb{R}_c^{\#n})$, it is sufficient to prove that $\mathcal{F}^{-1}(\mathcal{F}[f]) = f$ for $f$ contained in the dense set $C_0^{\infty}(*\mathbb{R}_c^{\#n})$. Choose $\varepsilon \approx 0$ infinite small enough so that the support of $f$ is contained in $C_\varepsilon$. Let $\mathbf{K}_\varepsilon = \{\mathbf{k} \in *\mathbb{R}_c^{\#n} | \text{ each } k_i/\varepsilon\pi^\# \in \mathbf{k} \text{ is an integer } \}$

$$ f(x) = \mathrm{Ext} - \sum_{\mathbf{k} \in \mathbf{K}_\varepsilon} \left( \left( \frac{1}{2}\varepsilon \right)^{n/2} [\mathrm{Ext} - \exp(i\mathbf{k} \cdot x)], f \right) \left( \frac{1}{2}\varepsilon \right)^{n/2} [\mathrm{Ext} - \exp(i\mathbf{k} \cdot x)] \quad (3.9) $$

is just the hyper infinite Fourier series of $f$ which #-converges uniformly in $C_\varepsilon$ to $f$ since $f$ is #-continuously #-differentiable. Thus

$$ f(x) = \mathrm{Ext} - \sum_{\mathbf{k} \in \mathbf{K}_\varepsilon} \frac{\hat{f}(k)[\mathrm{Ext} - \exp(i\mathbf{k} \cdot x)]}{(2\pi^\#)^{n/2}} (\varepsilon\pi^\#)^n. \quad (3.10) $$

Since $*\mathbb{R}_c^{\#n}$ is the disjoint union of the cubes of volume $(\varepsilon\pi^\#)^n$ centered about the points in $\mathbf{K}_\varepsilon$, the right-hand side of (1.10) is just a hyper finite Riemann sum for the integral of the function $\hat{f}(k)[\mathrm{Ext} - \exp(i\mathbf{k} \cdot x)]$. By the lemma 3.1, $\hat{f}(k)[\mathrm{Ext} - \exp(i\mathbf{k} \cdot x)] \in S^#(*\mathbb{R}_c^{\#n})$, so the hyperfinite Riemann sums (1.10) #-converge to the integral. Thus $\mathcal{F}^{-1}(\mathcal{F}[f]) = f$.
**Corollary 3.1.** Suppose $f \in S^#(*\mathbb{R}_c^{\#n})$. Then
$$ \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} |f(x)|^2 d^{\#n}x = \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} |f(k)|^2 d^{\#n}k. \qquad (3.11) $$

**Proof.** This is really a corollary of the proof rather than the statement of Theorem 1.1. If $f$ has #-compact support, then for $\varepsilon \approx 0$ small enough,

$$ f(x) = \mathrm{Ext-} \sum_{\mathbf{k} \in \mathbb{K}_{\varepsilon}} \left( \left( \frac{1}{2} \varepsilon \right)^{n/2} [\mathrm{Ext-} \exp(i\mathbf{k} \cdot x)], f \right) \left( \frac{1}{2} \varepsilon \right)^{n/2} [\mathrm{Ext-} \exp(i\mathbf{k} \cdot x)] \quad (3.12) $$

Since $\{ (\frac{1}{2}\varepsilon)^{n/2n/2} [\mathrm{Ext-} \exp(i\mathbf{k} \cdot x)] \}_{\mathbf{k} \in \mathbb{K}_{\varepsilon}}$ is an orthonormal basis for $\mathcal{L}_2^{\#}(C_{\varepsilon})$,

$$ \begin{aligned} \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} |f(x)|^2 d^{\#n}x = \mathrm{Ext-} \int_{C_{\varepsilon}} |f(x)|^2 d^{\#n}x = & \sum_{\mathbf{k} \in \mathbb{K}_{\varepsilon}} \left| \left(\frac{1}{2}\varepsilon\right)^{n/2} ([\mathrm{Ext-} \exp(i\mathbf{k} \cdot x)], f(x)) \right|^2 \\ & \sum_{\mathbf{k} \in \mathbb{K}_{\varepsilon}} |\hat{f}(k)|^2 (\varepsilon\pi_{\#})^n \underset{\varepsilon \to 0}{\to} \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} |f(k)|^2 d^{\#n}k. \end{aligned} \qquad (3.13) $$

This proves the corollary for $f \in C_0^{\infty}(*\mathbb{R}_c^{\#n})$. Since $f \mapsto \hat{f}$ and $\|\cdot\|_2$ are #-continuous on $S^{\#}(*\mathbb{R}_c^{\#n})$ and $C_0^{\infty}(*\mathbb{R}_c^{\#n})$ is #-dense, the result holds for all of $S^{\#}(*\mathbb{R}_c^{\#n})$.

**Definition 3.2.** Let $T \in S^{\#}(*\mathbb{R}_c^{\#n})$ the Fourier transform of $T$, denoted by $\hat{T}$ or $\mathcal{F}[T]$, is the tempered distribution defined by $\hat{T}(\varphi) = T(\hat{\varphi})$.

Suppose that $h, \varphi \in S^{\#}(*\mathbb{R}_c^{\#n})$, then by the polarization identity and the corollary to Theorem 1.1 we have $(h, \varphi) = (\hat{h}, \hat{\varphi})$. Substituting $\overline{\mathcal{F}[g]} = \mathcal{F}^{-1}[\overline{g}]$ for $h$, we obtain

$$ T_{\hat{g}}(\varphi) = \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} \hat{g}(x)\varphi(x)d^{\#n}x = \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} g(x)\hat{\varphi}(x)d^{\#n}x = T_g(\hat{\varphi}) = \hat{T}_g(\varphi). $$

where $T_{\hat{g}}$ and $T_g$ are the distributions corresponding to the functions $\hat{g}$ and $g$ respectively. This shows that the Fourier transform on $S^{\#}(*\mathbb{R}_c^{\#n})$ extends the transform we previously defined on $S^{\#}(*\mathbb{R}_c^{\#n})$.

**Theorem 3.2.** The Fourier transform is a one-to-one linear bijection from $S^{\#}(*\mathbb{R}_c^{\#n})$ to $S^{\#}(*\mathbb{R}_c^{\#n})$ which is the unique weakly #-continuous extension of the Fourier transform on $S^{\#}(*\mathbb{R}_c^{\#n})$.

**Proof.** If hyper infinite sequence $\{\varphi_n\}_{n \in \mathbb{N}^\#}$ #-convergence to $\varphi \in S^\#$, then by Theorem 1.1, hyper infinite sequence $\{\hat{\varphi}_n\}_{n \in \mathbb{N}^\#}$ #-convergence to $\hat{\varphi} \in S^\#$, so $T(\hat{\varphi}_n) \to_{\#} T(\hat{\varphi})$ for each $T \in S^{\#}$. Thus #-lim$_{n \to \infty}$ $T(\hat{\varphi}_n) = T(\hat{\varphi})$, which shows that $T$ is a #-continuous linear functional on $S^\#$. Furthermore, if $T_n \xrightarrow{w} T$, then $\hat{T}_n \xrightarrow{w} \hat{T}$ because $T(\hat{\varphi}_n) \to_{\#} T(\hat{\varphi})$ implies $\hat{T}(\varphi_n) \to_{\#} \hat{T}(\varphi)$. Thus $T \mapsto \hat{T}$ is weakly #-continuous.

**Definition 3.3.** Suppose that $f, g \in S^{\#}(*\mathbb{R}_c^{\#n})$. Then the convolution of $f$ and $g$, denoted by $f * g$, is the function

$$ (f * g)(y) = \mathrm{Ext-} \int_{*\mathbb{R}_c^{\#n}} f(y-x)g(x)d^{\#n}x. \qquad (3.14) $$

Convolutions frequently occur when one uses the Fourier transform because the Fourier transform takes products into convolutions.

**Theorem 3.3.** (a) For each $f \in S^{\#}(*\mathbb{R}_c^{\#n})$, $g \mapsto f * g$ is a #-continuous map of $S^{\#}(*\mathbb{R}_c^{\#n})$ into $S^{\#}(*\mathbb{R}_c^{\#n})$.

$$ (b) \quad \hat{fg} = (2\pi\#)^{-n/2} \hat{f} * \hat{g} \quad \text{and} \quad \widehat{f * g} = (2\pi\#)^{n/2} \hat{f}\hat{g}. $$
(c) For $f,g,h \in S^{#}(*\mathbb{R}_c^{\#n})$, $f \ast g = g \ast f$ and $f \ast (g \ast h) = (f \ast g) \ast h$.

**Definition 3.4.** Suppose that $f \in S^{#}(*\mathbb{R}_c^{\#n})$, $T \in S^{#}(*\mathbb{R}_c^{\#n})$ and let $\tilde{f}(x)$ denote the function, $f(-x)$. Then, the convolution of $T$ and $f$ denoted $T \ast f$ is the distribution in $S^{#}(*\mathbb{R}_c^{\#n})$ given by $(T \ast f)(\varphi) = T(\tilde{f} \ast \varphi)$ for all $\varphi \in S^{#}(*\mathbb{R}_c^{\#n})$.

The fact that $g \to \tilde{f} \ast g$ is a #-continuous transformation guarantees that $T \ast f \in S^{#}(*\mathbb{R}_c^{\#n})$.

**Abbreviation 3.1.** Let $f_y$ denote the function $f_y(x) = f(x - y)$ and $\tilde{f}_y$ the function $f(y - x)$. When $f$ is given by a longe expression $(\cdots)$, we will sometimes write $(\cdots)^{\sim}$ rather than $(\sim \cdot)$.

**Theorem 3.4.** For each $f \in S^{#}(*\mathbb{R}_c^{\#n})$ the map $T \to T \ast f$ is a weakly #-continuous map of $S^{#}(*\mathbb{R}_c^{\#n})$ into $S^{#}(*\mathbb{R}_c^{\#n})$ which extends the convolution on $S^{#}(*\mathbb{R}_c^{\#n})$. Furthermore,

(a) $T \ast f$ is a polynomially bounded $C^{\infty}$ function. In fact, $(T \ast f)(y) = T(\tilde{f}_y)$ and
$D^{\#\beta}(T \ast f) = (D^{\#\beta}T) \ast f = T \ast D^{\#\beta}f$;

(b) $(T \ast f) \ast g = T \ast (f \ast g)$;

(c) $\hat{T} \ast f = (2\pi_{\#})^{n/2} \hat{f} \hat{T}$.

**Theorem 3.5.** Let $T \in S^{#}(*\mathbb{R}_c^{\#n})$ and $f \in S^{#}(*\mathbb{R}_c^{\#n})$. Then $\hat{f}T \in O_M^n$ and $\hat{f}T(k) = (2\pi_{\#})^{n/2}T(f[\text{Ext}-\exp(-ik \cdot x)])$. In particular, if $T$ has #-compact support and $\psi \in S^{#}(*\mathbb{R}_c^{\#n})$ is identically one on a #-neighborhood of the support of $T$, then

$$ \hat{T}(k) = (2\pi_{\#})^{n/2} T(\psi[\text{Ext}-\exp(-ik \cdot x)]). \quad (3.15) $$

**Proof** By Theorem 3.4.c and the Fourier inversion formula we have

$$ \hat{f}T = (2\pi_{\#})^{n/2} \hat{f} \ast \hat{T}. \text{ Thus } \hat{f}T \in O_M^n \text{ and } \hat{f}T(k) = (2\pi_{\#})^{n/2} \hat{T}(\hat{f}_k) = (2\pi_{\#})^{n/2} T(f[\text{Ext}-\exp(-ik \cdot x)]) $$

**Remark 3.1.** We remark that one can also define the convolution of a distribution $T \in D^{#}(*\mathbb{R}_c^{\#n})$ with an $f \in D^{#}(*\mathbb{R}_c^{\#n})$ by $(T \ast f)(y) = T(\tilde{f}_y)$.

**Definition 3.5.** Let $j(x)$ be a positive $C^{\infty}$ function whose support lies in the sphere of radius one about the origin in $*\mathbb{R}_c^{\#n}$ and which satisfies $\text{Ext}-\int_{*\mathbb{R}_c^{\#n}} j(x) d^{\#n}x = 1$. The function $j_{\varepsilon}(x) = \varepsilon^{-n}j(x/\varepsilon)$, $\varepsilon \approx 0$ is called an approximate identity.

**Proposition 3.1.** Suppose $T \in S^{#}(*\mathbb{R}_c^{\#n})$ and let $j_{\varepsilon}(x)$ be an approximate identity. Then $T \ast j_{\varepsilon}(x) \to_{\#} T$ weakly as $\varepsilon \to_{\#} 0$.

**Proof.** If $\varphi \in S^{#}(*\mathbb{R}_c^{\#n})$, then $(T \ast j_{\varepsilon})(\varphi) = T(\tilde{j}_{\varepsilon} \ast \varphi)$, so it is sufficient to show that $\tilde{j}_{\varepsilon} \ast \varphi \to_{\#} \varphi$ in $S^{#}(*\mathbb{R}_c^{\#n})$. To do this it is sufficient to show that $(2\pi_{\#})^{n/2} \tilde{j}_{\varepsilon} \hat{\varphi} \to_{\#} \hat{\varphi}$ in $S^{#}(*\mathbb{R}_c^{\#n})$. Since $\tilde{j}_{\varepsilon}(\lambda) = j(\varepsilon\lambda)$ and $j(0) = (2\pi_{\#})^{n/2}$, it follows that $(2\pi_{\#})^{n/2} \tilde{j}_{\varepsilon}(x)$ #-converges to 1 uniformly on #-compact sets and is uniformly bounded. Similarly, $D^{\#\alpha} \tilde{j}_{\varepsilon}$ #-converges uniformly to zero. We conclude that $(2\pi_{\#})^{n/2} \tilde{j}_{\varepsilon} \hat{\varphi} \to_{\#} \hat{\varphi}$.

**Theorem 3.6** (The generalized Plancherel theorem) The Fourier transform extends uniquely to a unitary map of $L_2^{\#}(*\mathbb{R}_c^{\#n})$ onto $L_2^{\#}(*\mathbb{R}_c^{\#n})$. The inverse transform extends uniquely to its adjoint.

**Proof** The corollary to Theorem 3.1 states that if $f \in S^{#}(*\mathbb{R}_c^{\#n})$, then $\|f\|_2 = \|\hat{f}\|_2$. Since $\mathcal{F}[S^{#}] = S^{#}$ is a surjective isometry on $L_2^{\#}(*\mathbb{R}_c^{\#n})$.

**Theorem 3.7** (The generalized Riemann-Lebesgue lemma) The Fourier transform
extends uniquely to a bounded map from $\mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ into $C^{\infty}(*\mathbb{R}_c^{\#n})$, the $\#$-continuous functions vanishing at $\infty^{\#}$.

**Proof** For $f \in S^{\#}(*\mathbb{R}_c^{\#n})$, we know that $\hat{f} \in S^{\#}(*\mathbb{R}_c^{\#n})$ and thus $\hat{f} \in C^{\infty}(*\mathbb{R}_c^{\#n})$. The estimate is trivial. The Fourier transform is thus a bounded linear map from a $\#$-dense set of $\mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ into $C^{\infty}(*\mathbb{R}_c^{\#n})$. By the generalized B.L.T. theorem, extends uniquely to a bounded linear transformation of $C^{\infty}(*\mathbb{R}_c^{\#n})$ into $C^{\infty}(*\mathbb{R}_c^{\#n})$.

**Remark 3.2.** We remark that the Fourier transform takes $\mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ into, but not onto $C^{\infty}(*\mathbb{R}_c^{\#n})$.

A simple argument with test functions shows that the extended transform on $\mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ and $\mathcal{L}_2^{\#}(*\mathbb{R}_c^{\#n})$ is the restriction of the transform on $S^{\#}(*\mathbb{R}_c^{\#n})$, but it is useful to have an explicit integral representation. For $f \in \mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$, this is easy since we can find $f_m \in S^{\#}(*\mathbb{R}_c^{\#n})$ so that $\#\lim_{m \to \infty^{\#}} \|f - f_m\|_1 = 0$. Then, for each $\lambda$,

$$f(\lambda) = \#\lim_{m \to \infty^{\#}} (\hat{f}_m(\lambda)) = \\ \#\lim_{m \to \infty^{\#}} \left\{ \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} [\mathrm{Ext}\text{-exp}(-ik \cdot x)]f_m(x)d^\#x \right) \right\} = \\ \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{*\mathbb{R}_c^{\#n}} [\mathrm{Ext}\text{-exp}(-ik \cdot x)]f(x)d^\#x \right). \quad (3.16)$$

So, the Fourier transform of a function in $\mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ is given by the usual formula. Next, suppose $f \in \mathcal{L}_2^{\#}(*\mathbb{R}_c^{\#n})$ and let

$$\chi_R(x) = \begin{cases} 1 & \text{if } |x| \le R \\ 0 & \text{if } |x| > R \end{cases} \quad (3.17)$$

Then $\chi_R f \in \mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n})$ and $\#\lim_{R \to \infty^{\#}} \chi_R f = f$ in $\mathcal{L}_2^{\#}$, so by the generalized Plancherel theorem $\#\lim_{R \to \infty^{\#}} \chi_R f = \hat{f}$ in $\mathcal{L}_2^{\#}$. Thus

$$f(\lambda) = \#\lim_{R \to \infty^{\#}} \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int_{|x| \le R} [\mathrm{Ext}\text{-exp}(-ik \cdot x)]f(x)d^\#x \right) \quad (3.18)$$

where by $\#\lim_{R \to \infty^{\#}}$ we mean the $\#$-limit in the $\mathcal{L}_2^{\#}$-norm. Sometimes we will dispense with $|x| \le R$ and just write

$$f(\lambda) = \#\lim_{R \to \infty^{\#}} \frac{1}{(2\pi^\#)^{n/2}} \left( \mathrm{Ext} - \int [\mathrm{Ext}\text{-exp}(-ik \cdot x)]f(x)d^\#x \right) \quad (3.19)$$

for functions $f \in \mathcal{L}_2^{\#}(*\mathbb{R}_c^{\#n})$.

We have proven above that $\mathcal{F}: \mathcal{L}_2^{\#}(*\mathbb{R}_c^{\#n}) \to \mathcal{L}_2^{\#}(*\mathbb{R}_c^{\#n})$ and $\mathcal{F}: \mathcal{L}_1^{\#}(*\mathbb{R}_c^{\#n}) \to \mathcal{L}_{\infty^{\#}}^{\#}(*\mathbb{R}_c^{\#n})$ and in both cases is a bounded operator.

**Theorem 3.8 (Generalized Hausdorff-Young inequality)** Suppose $1 \le q \le 2$, and $p^{-1} + q^{-1} = 1$. Then the Fourier transform is a bounded map of $\mathcal{L}_p^{\#}(*\mathbb{R}_c^{\#n})$ to $\mathcal{L}_q^{\#}(*\mathbb{R}_c^{\#n})$ and its norm is less than or equal to $(2\pi^\#)^{n(1/2-1/q)}$.

# Chapter III. Non-Archiedean Hilbert Spaces over field $\#\mathbb{C}^\#$.

## § 1. Non-Archiedean Hilbert Spaces over field $\#\mathbb{C}^\#$
# Basics.

**Definition 1.1.** Let $H$ be external hyper infinite dimensional vector space over field $*\mathbb{C}_c^\# = *\mathbb{R}_c^\# + i*\mathbb{R}_c^\#$. An inner \#-product(or non-Archiedean inner product) on $H$ is a $*\mathbb{C}_c^\#$-valued function, $\langle\cdot, \cdot\rangle_\# : H \times H \to *\mathbb{C}_c$, such that

$$ (1) \quad \langle ax + by, z \rangle_\# = a\langle x, z \rangle + b\langle y, z \rangle_\# \text{, i.e. } x \to \langle x, z \rangle_\# \text{ is linear.} $$

$$ (2) \quad \overline{\langle x, y \rangle}_\# = \langle y, x \rangle_\# $$

$$ (3) \quad \|x\|_\#^2 = \langle x, x \rangle_\# \ge 0 \text{ with equality } \|x\|_\#^2 = 0 \text{ iff } x = 0. $$

Notice that combining properties (1) and (2) that $x \to \langle z, x \rangle$ is anti-linear for fixed $z \in H$, i.e. $\langle z, ax + by \rangle_\# = \bar{a}\langle z, x \rangle_\# + \bar{b}\langle z, y \rangle_\#$.

The following formula useful:

$$ \|x+y\|_{\#}^{2} = \langle x+y, x+y \rangle_{\#} = \|x\|_{\#}^{2} + \|y\|_{\#}^{2} + \langle x,y \rangle_{\#} + \langle y,x \rangle_{\#} = \|x\|_{\#}^{2} + \|y\|_{\#}^{2} + 2 \operatorname{Re}\langle x,y \rangle_{\#} \quad (1.1) $$

**Theorem 1.1.** (Generalized Schwarz Inequality). Let $(H, \langle\cdot, \cdot\rangle_\#)$ be an inner \#-product space, then for all $x, y \in H$

$$ |\langle x, y \rangle_\#| \le \|x\|_\# \|y\|_\# \quad (1.2) $$

and equality holds iff $x$ and $y$ are linearly dependent.

**Proof.** If $y = 0$, the result holds trivially. So assume that $y \ne 0$. First off notice that if $x = \alpha y$ for some $\alpha \in *\mathbb{C}_c^\#$, then $\langle x, y \rangle = \alpha \|y\|_\#^2$ and hence $|\langle x, y \rangle_\#| = |\alpha \|y\|_\#^2 = \|x\|_\# \|y\|_\#$. Note that in this case $\alpha = \langle x, y \rangle \|y\|^2$. Now suppose that $x \in H$ is arbitrary, let $z = x - \|y\|_\#^{-2} \langle x, y \rangle_\# y$. So $z$ is the orthogonal projection of $x$ onto $y$. Then

$$ \begin{aligned} 0 \le \|z\|_{\#}^{2} &= \left\| x - \frac{\langle x, y \rangle_{\#}}{\|y\|_{\#}^{2}} y \right\|_{\#}^{2} = \|x\|_{\#}^{2} + \frac{|\langle x, y \rangle_{\#}|^{2}}{\|y\|_{\#}^{4}} \|y\|_{\#}^{2} - 2 \operatorname{Re} \left\langle x, \frac{\langle x, y \rangle_{\#}}{\|y\|_{\#}^{2}} y \right\rangle = \\ &= \|x\|_{\#}^{2} - \frac{|\langle x, y \rangle_{\#}|^{2}}{\|y\|_{\#}^{2}}. \end{aligned} \quad (1.3) $$

from (1.3) it follows that $0 \le \|y\|_\#^2 \|x\|_\#^2 - |\langle x, y \rangle_\#|^2$ with equality iff $z = 0$ or equivalently iff $x = \|y\|_\#^{-2} \langle x, y \rangle_\# y$.

**Corollary 1.1.** Let $(H, \langle\cdot, \cdot\rangle)$ be an inner \#-product space and $\|x\|_\# := \sqrt{\langle x, x \rangle_\#}$. Then $\|\cdot\|_\#$ is a $*\mathbb{R}_c^\#$-valued \#-norm on $H$. Moreover $\langle\cdot, \cdot\rangle_\#$ is \#-continuous on $H \times H$, where $H$ is viewed as the \#-normed space $(H, \|\cdot\|_\#)$.

**Proof.** The only non-trivial thing to verify that $\|\cdot\|_\#$ is a \#-norm is the triangle inequality:

$$ \|x+y\|^2 = \|x\|^2 + \|y\|^2 + 2 \operatorname{Re} \langle x, y \rangle_\# \le \|x\|^2 + \|y\|^2 + 2 \|x\|_\# \|y\|_\# = (\|x\|_\# + \|y\|_\#)^2 $$
where we have made use of Schwarz's inequality. Taking the square root of this inequality shows $\|x+y\| \le \|x\| + \|y\|$. For the \#-continuity assertion:

$$ \begin{aligned} |\langle x, y \rangle_\# - \langle x', y' \rangle_\#| &= |\langle x - x', y \rangle_\# + \langle x', y - y' \rangle_\#| \le \|y\|_\# \|x - x'\|_\# + \|x'\|_\# \|y - y'\|_\# \\ &\le \|y\|_\# \|x - x'\|_\# + (\|x\|_\# + \|x - x'\|_\#) \|y - y'\|_\# = \|y\|_\# \|x - x'\|_\# + \|x\|_\# \|y - y'\|_\# \\ &\quad + \|x - x'\|_\# \|y - y'\|_\# \text{ from which it follows that } \langle\cdot, \cdot\rangle \text{ is \#-continuous.} \end{aligned} $$

**Definition 1.2.** Let $(H, \langle\cdot, \cdot\rangle_\#)$ be an inner \#-product space, we say $x, y \in H$ are orthogonal and write $x \perp y$ iff $\langle x, y \rangle_\# = 0$. More generally if $A \subset H$ is a set, $x \in H$ is orthogonal to $A$ and write $x \perp A$ iff $\langle x, y \rangle = 0$ for all $y \in A$. Let $A_\perp = \{x \in H : x \perp A\}$ be the set of vectors orthogonal to $A$. We also say that a set $S \subset H$ is orthogonal if $x \perp y$ for all $x, y \in S$ such that $x \ne y$. If $S$ further