hcju/proofwikips huggingface dataset:
Fatou's Lemma for Integrals/Positive Measurable Functions
Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$, $f_n: X \to \overline \R$ be a sequence of positive measurable functions. Let $\displaystyle \liminf_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit inferior of the $f_n$. Then: :$\displaystyle \int \liminf_{n \mathop \to \infty} f_n \rd \mu \le \liminf_{n \mathop \to \infty} \int f_n \rd \mu$ where: :the integral sign denotes $\mu$-integration :the  limit inferior is taken in the extended real numbers $\overline \R$.
Reverse Fatou's Lemma/Integrable Functions
Let $\sequence {f_n}_{n \mathop \in \N} \in \LL^1$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions. Let $\displaystyle \limsup_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit superior of the $f_n$. Suppose that there exists an $\mu$-integrable $f: X \to \R$ such that for all $n \in \N$, $f_n \le f$ pointwise. Then: :$\displaystyle \limsup_{n \mathop \to \infty} \int f_n \rd \mu \le \int \limsup_{n \mathop \to \infty} f_n \rd \mu$ where: :the integral sign denotes $\mu$-integration :the  limit inferior is taken in the extended real numbers $\overline \R$.
Set is Subset of Union/Family of Sets
Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$. Then: :$\displaystyle \forall \beta \in I: S_\beta \subseteq \bigcup_{\alpha \mathop \in I} S_\alpha$ where $\displaystyle \bigcup_{\alpha \mathop \in I} S_\alpha$ is the union of $\family {S_\alpha}$.
Reverse Fatou's Lemma/Positive Measurable Functions
Let $\sequence {f_n}_{n \mathop \in \N} \in \MM_{\overline \R}^+$, $f_n: X \to \overline \R$ be a sequence of positive measurable functions. Suppose that there exists a positive measurable function $f: X \to \overline \R$ such that: :$\displaystyle \int f \rd \mu < +\infty$ :$\forall n \in \N: f_n \le f$ where $\le$ signifies a pointwise inequality. Let $\displaystyle \limsup_{n \mathop \to \infty} f_n: X \to \overline \R$ be the pointwise limit superior of the $f_n$. Then: :$\displaystyle \limsup_{n \mathop \to \infty} \int f_n \rd \mu \le \int \limsup_{n \mathop \to \infty} f_n \rd \mu$ where: :the integral sign denotes $\mu$-integration :the  limit superior is taken in the extended real numbers $\overline \R$.
Monotone Convergence Theorem (Measure Theory)
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {u_n}_{n \mathop \in \N} \in \map {\LL^1} \mu$, $u_n: X \to \R$ be a increasing sequence of $\mu$-integrable functions. Let $\displaystyle \sup_{n \mathop \in \N} u_n: X \to \overline \R$ be the pointwise supremum of the $u_n$. Then $\displaystyle \sup_{n \mathop \in \N} u_n$ is $\mu$-integrable : :$\displaystyle \sup_{n \mathop \in \N} \int u_n \rd \mu < +\infty$ and, in that case: :$\displaystyle \int \sup_{n \mathop \in \N} u_n \rd \mu = \sup_{n \mathop \in \N} \int u_n \rd \mu$
Intersection is Associative/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ and $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of sets. Let $\displaystyle I = \bigcap_{\lambda \mathop \in \Lambda} I_\lambda$. Then: :$\displaystyle \bigcap_{i \mathop \in I} S_i = \bigcap_{\lambda \mathop \in \Lambda} \paren {\bigcap_{i \mathop \in I_\lambda} S_i}$
Union is Associative/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ and $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of sets. Let $\displaystyle I = \bigcup_{\lambda \mathop \in \Lambda} I_\lambda$ denote the union of  $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$. Then: :$\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} S_i}$
Lebesgue's Dominated Convergence Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^1} \mu$, $f_n: X \to \R$ be a sequence of $\mu$-integrable functions. Suppose that for some $\mu$-integrable $g: X \to \R$, it holds that: :$\forall n \in \N: \size {f_n} \le g$ pointwise Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists almost everywhere. Then $f$ is $\mu$-integrable, and: :$\displaystyle \lim_{n \mathop \to \infty} \int \size {f_n - f} \rd \mu = 0$ :$\displaystyle \lim_{n \mathop \to \infty} \int f_n \rd \mu = \int \lim_{n \mathop \to \infty} f_n \rd \mu$
Continuity under Integral Sign
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $U$ be a non-empty open set of a metric space. Let $f: U \times X \to \R$ be a mapping satisfying: :$(1): \quad$ For all $\lambda \in U$, the mapping $x \mapsto \map f {\lambda, x}$ is $\mu$-integrable :$(2): \quad$ For $\mu$-almost all $x \in X$, the mapping $\lambda \mapsto \map f {\lambda, x}$ is continuous :$(3): \quad$ There exists a $\mu$-integrable $g: X \to \R$ such that: ::::$\forall \tuple {\lambda, x} \in U \times X: \size {\map f {\lambda, x} } \le \map g x$ Then the mapping $h: U \to \R$ defined by: :$\displaystyle \map h \lambda := \int \map f {\lambda, x} \map {\rd \mu} x$ is continuous.
Differentiability under Integral Sign
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\openint a b$ be a non-empty open interval. Let $f: \openint a b \times X \to \R$ be a mapping satisfying: :$(1): \quad$ For all $t \in \openint a b$, the mapping $x \mapsto f \left({t, x}\right)$ is $\mu$-integrable :$(2): \quad$ For all $x \in X$, the mapping $t \mapsto \map f {t, x}$ is differentiable :$(3): \quad$ There exists a $\mu$-integrable $g: X \to \R$ such that: ::::$\displaystyle \forall \tuple {t, x} \in \openint a b \times X: \size {\frac \partial {\partial t} \map f {t, x} } \le \map g x$ Then the mapping $h: \openint a b \to \R$ defined by: :$\displaystyle \map h t := \int \map f {t, x} \map {\rd \mu} x$ is differentiable, and its derivative is: :$\displaystyle D_t \map h t = \int \frac \partial {\partial t} \map f {t, x} \map {\rd \mu} x$
Riemann-Lebesgue Theorem
Let $f: \closedint a b \to \R$ be a bounded mapping. Let $\mu$ be a one-dimensional Lebesgue measure. {{explain|Link to definition of one-dimensional in this specific context}} Then $f$ is Darboux integrable  the set of all discontinuities of $f$ is a $\mu$-null set.
Gamma Function is Continuous on Positive Reals
Let $\Gamma: \R_{>0} \to \R$ be the Gamma function, restricted to the strictly positive real numbers. Then $\Gamma$ is continuous.
Gamma Function is Smooth on Positive Reals
Let $\Gamma: \R_{>0} \to \R$ be the Gamma function, restricted to the strictly positive real numbers. Then $\Gamma$ is smooth. {{expand|I'm quite sure it's analytic on all of its domain}}
Log of Gamma Function is Convex on Positive Reals
Let $\Gamma: \R_{>0} \to \R$ be the Gamma function, restricted to the strictly positive real numbers. Let $\ln$ denote the natural logarithm function. Then the composite mapping $\ln \circ \operatorname \Gamma$ is a convex function.
Pratt's Lemma
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let: :$\sequence {g_n}_{n \mathop \in \N}$ :$\sequence {G_n}_{n \mathop \in \N}$ :$\sequence {f_n}_{n \mathop \in \N}$ be sequences of $\mu$-integrable functions. Let the pointwise limits: :$\displaystyle f := \lim_{n \mathop \to \infty} f_n$ :$\displaystyle g := \lim_{n \mathop \to \infty} g_n$ :$\displaystyle G := \lim_{n \mathop \to \infty} G_n$ exist. Let $g$ and $G$ be $\mu$-integrable. Suppose that, for all $x \in X$ and $n \in \N$: :$\map {g_n} x \le \map {f_n} x \le \map {G_n} x$ Finally, suppose the following hold: :$\displaystyle \lim_{n \mathop \to \infty} \int g_n \rd \mu = \int g \rd \mu$ :$\displaystyle \lim_{n \mathop \to \infty} \int G_n \rd \mu = \int G \rd \mu$ Then: :$\displaystyle \lim_{n \mathop \to \infty} \int f_n \rd \mu = \int f \rd \mu$ and the latter is finite.
Hölder's Inequality
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p, q \in \R$ such that $\dfrac 1 p + \dfrac 1 q = 1$. Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space. Then their pointwise product $f g$ is $\mu$-integrable, that is: :$f g \in \map {\LL^1} \mu$ and: :$\size {f g}_1 = \displaystyle \int \size {f g} \rd \mu \le \norm f_p \cdot \norm g_q$ where the $\norm {\, \cdot \,}_p$ signify $p$-seminorms.
Hölder's Inequality/Equality
:$\displaystyle \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$ holds , for almost all $x \in X$: :$\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$
Condition for Membership of Equivalence Class
Let $\RR$ be an equivalence relation on a set $S$. Let $\eqclass x \RR$ denote the $\RR$-equivalence class of $x$. Then: :$\forall y \in S: y \in \eqclass x \RR \iff \tuple {x, y} \in \RR$
Cartesian Product is Set Product/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets. For all $j \in I$, let $\pr_i: \displaystyle \prod_{j \mathop \in I} \family {S_j} \to S_i$ be the $i$th projection from $\displaystyle \prod_{j \mathop \in I} \family {S_j}$ to $S_i$. Then $\struct {\displaystyle \prod_{j \mathop \in I} \family {S_j}, \family {\pr_i}_{i \mathop \in I} }$ is a set product.
Preimage of Intersection under Relation/General Result
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation. Let $\mathcal P \left({T}\right)$ be the power set of $T$. Let $\mathbb T \subseteq \mathcal P \left({T}\right)$. Then: :$\displaystyle \mathcal R^{-1} \left[{\bigcap \mathbb T}\right] \subseteq \bigcap_{X \mathop \in \mathbb T} \mathcal R^{-1} \left[{X}\right]$
Image of Union under Relation/Family of Sets
Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $\RR \subseteq S \times T$ be a relation. Then: :$\displaystyle \RR \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \RR \sqbrk {S_i}$ where $\displaystyle \bigcup_{i \mathop \in I} S_i$ denotes the union of $\family {S_i}_{i \mathop \in I}$.
Image of Intersection under Relation/Family of Sets
Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $\RR \subseteq S \times T$ be a relation. Then: :$\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}$ where $\ds \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\family {S_i}_{i \mathop \in I}$.
Preimage of Union under Relation/Family of Sets
Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Let $\RR \subseteq S \times T$ be a relation. Then: :$\displaystyle \RR^{-1} \sqbrk {\bigcup_{i \mathop \in I} T_i} = \bigcup_{i \mathop \in I} \RR^{-1} \sqbrk {T_i}$ where: :$\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\family {T_i}_{i \mathop \in I}$ :$\RR^{-1} \left[{T_i}\right]$ denotes the preimage of $T_i$ under $\RR$.
Image of Intersection under Mapping/Family of Sets
Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $f: S \to T$ be a mapping. Then: :$\ds f \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} f \sqbrk {S_i}$ where $\ds \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\family {S_i}_{i \mathop \in I}$.
Image of Union under Mapping/Family of Sets
Let $S$ and $T$ be sets. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Let $f: S \to T$ be a mapping. Then: :$\ds f \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} f \sqbrk {S_i}$ where $\ds \bigcup_{i \mathop \in I} S_i$ denotes the union of $\family {S_i}_{i \mathop \in I}$.
Minkowski's Inequality/Lebesgue Spaces
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R$, $p \ge 1$. Let $f, g: X \to \R$ be $p$-integrable, that is, elements of Lebesgue $p$-space $\map {\mathcal L^p} \mu$. Then their pointwise sum $f + g: X \to \R$ is also $p$-integrable, and: :$\norm {f + g}_p \le \norm f_p + \norm g_p$ where $\norm {\, \cdot \, }_p$ denotes the $p$-seminorm.
Lebesgue Space is Vector Space
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mathcal L^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$. Then $\mathcal L^p \left({\mu}\right)$ is a vector subspace of $\mathcal M \left({\Sigma}\right)$, the space of $\Sigma$-measurable functions. In particular, it is a vector space.
Triangle Inequality for Series/Lebesgue Spaces
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R$, $p \ge 1$. Let $\sequence {f_n}_{n \mathop \in \N} \in \map {\LL^p} {\mu}$ be a sequence of $p$-integrable functions, that is, a sequence in Lebesgue $p$-space. Suppose that for all $n \in \N$, $f_n \ge 0$ holds pointwise. {{expand|This condition could possibly be weakened/altered, using Lebesgue's Dominated Convergence Theorem in place of Beppo Levi's Theorem}} Then: :$\displaystyle \norm {\sum_{n \mathop = 1}^\infty f_n}_p \le \sum_{n \mathop = 1}^\infty \norm {f_n}_p$ where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.
Riesz-Fischer Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R$, $p \ge 1$. The Lebesgue $p$-space $\map {\LL^p} \mu$, endowed with the $p$-norm, is a complete metric space.
Pointwise Convergent Bounded Sequence in Lebesgue Space Converges in Norm
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R_{\ge 1}$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n: X \to \R$ be a sequence in Lebesgue $p$-space $\LL^p \left({\mu}\right)$. Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists $\mu$-almost everywhere. Suppose that for some $g \in \map {\LL^p} \mu$, the pointwise inequality $\size {f_n} \le g$ holds for all $n \in \N$. Then $f \in \map {\LL^p} \mu$, and: :$\displaystyle \lim_{n \mathop \to \infty} \norm {f - f_n}_p = 0$ where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.
Riesz's Convergence Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $p \in \R$, $p \ge 1$. Let $\sequence {f_n}_{n \mathop \in \N}, f_n: X \to \R$ be a sequence in Lebesgue $p$-space $\map {\LL^p} \mu$. Suppose that the pointwise limit $f := \displaystyle \lim_{n \mathop \to \infty} f_n$ exists $\mu$-almost everywhere, and that $f \in \map {\LL^p} \mu$. Then the following are equivalent: :$(1): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f - f_n}_p = 0$ :$(2): \quad \displaystyle \lim_{n \mathop \to \infty} \norm {f_n}_p = \norm f_p$ where $\norm {\, \cdot \,}_p$ denotes the $p$-seminorm.
Preimage of Union under Mapping/Family of Sets
Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Let $f: S \to T$ be a relation. Then: :$\displaystyle f^{-1} \sqbrk {\bigcup_{i \mathop \in I} T_i} = \bigcup_{i \mathop \in I} f^{-1} \sqbrk {T_i}$ where: :$\displaystyle \bigcup_{i \mathop \in I} T_i$ denotes the union of $\family {T_i}_{i \mathop \in I}$ :$f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.
Image of Intersection under One-to-Many Relation/Family of Sets
Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation. Then $\RR$ is a one-to-many relation : :$\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}$  where $\family {S_i}_{i \mathop \in I}$ is ''any'' family of subsets of $S$.
Preimage of Intersection under Mapping/Family of Sets
Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Let $f: S \to T$ be a mapping. Then: :$\displaystyle f^{-1} \sqbrk {\bigcap_{i \mathop \in I} T_i} = \bigcap_{i \mathop \in I} f^{-1} \sqbrk {T_i}$ where: :$\displaystyle \bigcap_{i \mathop \in I} T_i$ denotes the intersection of $\family {T_i}_{i \mathop \in I}$. :$f^{-1} \sqbrk {T_i}$ denotes the preimage of $T_i$ under $f$.
De Morgan's Laws (Set Theory)/Set Complement/General Case
Let $\mathbb T$ be a set of sets, all of which are subsets of a universe $\mathbb U$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Set Complement/General Case/Complement of Union}}
De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets
Let $S$ be a set. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Relative Complement/Family of Sets/Complement of Union}}
Simple P-Integrable Functions Dense in Lebesgue Space
Let $\left({X, \Sigma, \mu}\right)$ be a measure space, and let $p \in \R$, $p \ge 1$. Let $\mathcal L^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$. Let $\mathcal E \left({\Sigma}\right) \cap \mathcal L^p \left({\mu}\right)$ be the space of $\Sigma$-simple, $p$-integrable functions. Then $\mathcal E \left({\Sigma}\right) \cap \mathcal L^p \left({\mu}\right)$ is everywhere dense in $\mathcal L^p \left({\mu}\right)$.
De Morgan's Laws (Set Theory)/Set Difference/Family of Sets
Let $S$ and $T$ be sets. Let $\family {T_i}_{i \mathop \in I}$ be a family of subsets of $T$. Then: ==== Difference with Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Intersection}} ==== Difference with Union ==== {{:De Morgan's Laws (Set Theory)/Set Difference/Family of Sets/Difference with Union}}
De Morgan's Laws (Set Theory)/Set Complement/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets, all of which are subsets of a universe $\Bbb U$. Then: ==== Complement of Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection}} ==== Complement of Union ==== {{:De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union}}
Jensen's Inequality (Measure Theory)
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \R$ be a $\mu$-integrable function such that $f \ge 0$ pointwise.
Jensen's Inequality (Measure Theory)/Convex Functions
Let $V: \hointr 0 \infty \to \hointr 0 \infty$ be a convex function. Then for all positive measurable functions $g: X \to \R$, $g \in \map {\mathcal M^+} \Sigma$: :$\map V {\dfrac {\int g \cdot f \rd \mu} {\int f \rd \mu} } \le \dfrac {\int \paren {V \circ g} \cdot f \rd \mu} {\int f \rd \mu}$ where $\circ$ denotes composition, and $\cdot$ denotes pointwise multiplication.
Jensen's Inequality (Measure Theory)/Concave Functions
Let $\Lambda: \hointr 0 \infty \to \hointr 0 \infty$ be a concave function. Then for all positive measurable functions $g: X \to \R$, $g \in \map {\MM^+} \Sigma$: :$\dfrac {\int \paren {\Lambda \circ g} \cdot f \rd \mu} {\int f \rd \mu} \le \map \Lambda {\dfrac {\int g \cdot f \rd \mu} {\int f \rd \mu} }$ where $\circ$ denotes composition, and $\cdot$ denotes pointwise multiplication.
Power Set is Lattice
Let $S$ be a set. Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the subset relation $\subseteq$. Then $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is a lattice.
Divisor Relation induces Lattice
Let $\struct {\Z_{> 0}, \divides}$ be the ordered set comprising: :The set of positive integers $\Z_{> 0}$ :The divisor relation $\divides$ defined as: ::$a \divides b := \exists k \in \Z_{> 0}: b = ka$ Then $\struct {\Z_{> 0}, \divides}$ is a lattice.
Generator for Product Sigma-Algebra
Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces. Let $\GG_1$ and $\GG_2$ be generators for $\Sigma_1$ and $\Sigma_2$, respectively. Then $\GG_1 \times \GG_2$ is a generator for the product $\sigma$-algebra $\Sigma_1 \otimes \Sigma_2$.
Uniqueness of Product Measures
Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be measure spaces. Let $\GG_1$ and $\GG_2$ be generators for $\Sigma_1$ and $\Sigma_2$, respectively. Suppose that $\GG_1$ and $\GG_2$ are closed under intersection. Suppose further that there are exhausting sequences $\sequence {G_{1, n} }_{n \mathop \in \N}$ and $\sequence {G_{2, n} }_{n \mathop \in \N}$ in $\GG_1$ and $\GG_2$, respectively, such that: :$\forall n \in \N: \map \mu {G_{1, n} } < \infty$ :$\forall n \in \N: \map \nu {G_{2, n} } < \infty$ Then there is at most one measure $\rho$ on the product space $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2}$ such that: :$\forall G_1 \in \GG_1, G_2 \in \GG_2: \map \rho {G_1 \times G_2} = \map \mu {G_1} \, \map \nu {G_2}$ That is, there can be at most one product measure on $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2}$.
Existence of Product Measures
Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces. Then there exists a $\sigma$-finite measure $\rho$ on the product space $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ such that: :$\forall E_1 \in \Sigma_1, E_2 \in \Sigma_2: \rho \left({E_1 \times E_2}\right) = \mu \left({E_1}\right) \nu \left({E_2}\right)$ This $\rho$ is called product measure and is also denoted $\mu \times \nu$.
Tonelli's Theorem
Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be $\sigma$-finite measure spaces. Let $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$. Let $f: X \times Y \to \overline \R_{\ge 0}$ be a positive $\Sigma_1 \otimes \Sigma_2$-measurable function. Then: :$\displaystyle \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\d \mu} x \map {\d \nu} y = \int_X \int_Y \map f {x, y} \map {\d \nu} y \map {\d \mu} x$
Condition for Power Set to be Totally Ordered
Let $\mathcal P \left({S}\right)$ be the power set of a set $S$. Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the set $\mathcal P \left({S}\right)$ ordered by $\subseteq$. Then $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is totally ordered iff $S$ is either the empty set or a singleton.
Fubini's Theorem
Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be $\sigma$-finite measure spaces. Let $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$. Let $f: X \times Y \to \R$ be a $\Sigma_1 \otimes \Sigma_2$-measurable function. Suppose that: :$\displaystyle \int_{X \times Y} \size f \map {\rd} {\mu \times \nu} < \infty$ Then $f$ is $\mu \times \nu$-integrable, and: :$\displaystyle \int_{X \times Y} f \map {\rd} {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\rd \mu} x \map {\rd \nu} y = \int_X \int_Y \map f {x, y} \map {\rd \nu} y \map {\rd \mu} x$ {{expand|Implement a version of this for real functions}}
Product Sigma-Algebra Generated by Projections
Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces. Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X \times Y$. Let $\pr_1: X \times Y \to X$ and $\pr_2: X \times Y \to Y$ be the first and second projections, respectively. Then: :$\Sigma_1 \otimes \Sigma_2 = \map \sigma {\pr_1, \pr_2}$ where $\sigma$ denotes generated $\sigma$-algebra.
Measurable Mappings to Product Measurable Space
Let $\left({X, \Sigma}\right)$, $\left({X_1, \Sigma_1}\right)$ and $\left({X_2, \Sigma_2}\right)$ be measurable spaces. Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X_1 \times X_2$. Let $\operatorname{pr}_1: X_1 \times X_2 \to X_1$ and $\operatorname{pr}_2: X_1 \times X_2 \to X_2$ be the first and second projections, respectively. A mapping $f: X \to X_1 \times X_2$ is $\Sigma \, / \, \Sigma_1 \otimes \Sigma_2$-measurable : :$\operatorname{pr}_i \circ f: X \to X_i$ is $\Sigma \, / \, \Sigma_i$-measurable, for $i = 1, 2$
Measurable Mappings from Product Measurable Space
Let $\left({X, \Sigma}\right)$, $\left({X_1, \Sigma_1}\right)$ and $\left({X_2, \Sigma_2}\right)$ be measurable spaces. Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X_1 \times X_2$. Let $f: X_1 \times X_2 \to X$ be a $\Sigma_1 \otimes \Sigma_2 \, / \, \Sigma$-measurable mapping. Then: :$\forall x_1 \in X_1: f \left({x_1, \cdot}\right): X_2 \to X$ is $\Sigma_2 \, / \, \Sigma$-measurable :$\forall x_2 \in X_2: f \left({\cdot, x_2}\right): X_1 \to X$ is $\Sigma_1 \, / \, \Sigma$-measurable