Proof of Kit for def:expectation-variance

Throughout, formulas are written with abbreviations as in Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations (vi), a formula being predicative when it is so with its abbreviations eliminated; the given $x_{0},\dots,x_{N-1}$ are the first of distinct set variables $x_{0},x_{1},\dots$ as in Renaming and Substituting Variables in Predicative Formulas, so that $x_{0},\dots,x_{6}$ are distinct also when $N<7$; $u$, $w$, $v$, $y$ and $z$ are further distinct set variables, and the bound variables of the cited formulas are chosen other than all of these and the parameters; the earlier versions of the definitions on which Integrable Vector- and Matrix-Valued Functions and Their Integrals and Expectation are stated give the same measurable spaces, measures, Borel $\sigma$-algebras, measurable maps, sets $\mathcal{L}^{1}(P)$, Euclidean spaces and real matrices as the current ones (The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (i), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (ii), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (iv), The Earlier and the New Versions of the Definitions of σ-Algebras, Measures, Measurable Functions and Integrals Agree (vi), The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (i), The Earlier and the New Versions of the Definitions of Euclidean Space, Real Matrices and Normed Spaces Agree (ii)); and routine facts about maps and sets are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity (i), Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity (iii)), among them that $[n]$ is a set for $n\in\mathbb{N}$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i)).

The notions. Let $P$ be a measure on $(\Omega,\mathcal{F})$ such that $(\Omega,\mathcal{F},P)$ is a probability space, and let $n,p,q\in\mathbb{N}$. Then $(\Omega,\mathcal{F},P)$ is a measure space, $\mathcal{L}^{1}(P)$ is the set of integrable functions $\Omega\to\mathbb{R}$, and $\mathcal{L}^{1}(P;\mathbb{R}^{n})$ and $\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$ are the sets of integrable functions into $\mathbb{R}^{n}$ and into $\mathbb{R}^{p\times q}$, all formed for this measure space. The elements of $\mathcal{L}^{1}(P;\mathbb{R}^{n})$ are the $f\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ whose component maps satisfy

$$f_{c}\in\mathcal{L}^{1}(P)\qquad(c\in[n]),\tag{$*$}$$

and the elements of $\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$ are the $F\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$ whose entry maps satisfy

$$F_{ab}\in\mathcal{L}^{1}(P)\qquad(a\in[p],\ b\in[q]).\tag{$**$}$$

For a class $X$, $X$ is an integrable random variable if and only if $X\in\mathcal{L}^{1}(P)$. Indeed, by Expectation (ii), an integrable random variable is a random variable $X$ with $X\in\mathcal{L}^{1}(P)$; conversely, every $X\in\mathcal{L}^{1}(P)$ lies in $\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ (Integrable Functions and Their Lebesgue Integral (ii)), so it is a map $X:\Omega\to\mathbb{R}$ that is measurable as a map into $\mathbb{R}$ (Measurable Map, and the Set of Measurable Maps (ii)), that is, a random variable. In the same way, for a class $X$, $X$ is an integrable random vector in $\mathbb{R}^{n}$ if and only if $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$: by Expectation (iii), an integrable random vector is a random vector $X$ with $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$; conversely, every element of $\mathcal{L}^{1}(P;\mathbb{R}^{n})$ is a map $X:\Omega\to\mathbb{R}^{n}$ that is measurable as a map into $\mathbb{R}^{n}$ (Measurable Map, and the Set of Measurable Maps (ii)), that is, a random vector in $\mathbb{R}^{n}$. The set $\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$ is the one on which Expectation (iv) defines the expectation.

The formula for probability spaces. By Kit for def:probability-space-random-variable (i), with its $N$ equal to $3$ and $i=0$, $j=1$, $k=2$, there is a predicative formula $\pi$ in $x_{0},x_{1},x_{2}$ which, for sets $Y$, $\mathcal{G}$ and $m$ assigned to $x_{0}$, $x_{1}$ and $x_{2}$, holds if and only if $(Y,\mathcal{G})$ is a measurable space, $m$ is a measure on $(Y,\mathcal{G})$ and $(Y,\mathcal{G},m)$ is a probability space. By Renaming and Substituting Variables in Predicative Formulas (iii), with its $k$ equal to $3$, its $n$ equal to $1$ and $\tau_{0}=\Omega$, $\tau_{1}=\mathcal{F}$, $\tau_{2}=x_{0}$, the formula $\psi=\pi(\Omega,\mathcal{F},x_{0})$ is a predicative formula in $x_{0}$. As $(\Omega,\mathcal{F})$ is a measurable space, for a set $P$ assigned to $x_{0}$, $\psi$ holds if and only if $P$ is a measure on $(\Omega,\mathcal{F})$ and $(\Omega,\mathcal{F},P)$ is a probability space. By Renaming and Substituting Variables in Predicative Formulas (ii), $\psi$ is also a predicative formula in $x_{0},\dots,x_{m-1}$ for every $m\ge1$.

The formula for integrable functions. Let $\delta$ be the predicative formula in $x_{0},x_{1}$ given by Kit for def:lebesgue-integral-integrable (i), for the measurable space $(\Omega,\mathcal{F})$ fixed here, with its $N$ equal to $2$ and $i=0$, $j=1$. For sets $m$ and $f$ assigned to $x_{0}$ and $x_{1}$, $\delta$ holds if and only if $m$ is a measure on $(\Omega,\mathcal{F})$ and $f\in\mathcal{L}^{1}(m)$. By Renaming and Substituting Variables in Predicative Formulas (i), with its number of variables equal to $2$, its $n$ equal to $7$, $\sigma(0)=0$ and $\sigma(1)=4$, respectively $\sigma(1)=6$, the formulas $\delta(x_{0},x_{4})$ and $\delta(x_{0},x_{6})$ are predicative formulas in $x_{0},\dots,x_{6}$ which hold exactly when $\delta$ holds with the sets assigned to $x_{0}$ and $x_{4}$, respectively to $x_{0}$ and $x_{6}$, assigned to $x_{0}$ and $x_{1}$.

Integrable random variables. Let

$$\eta=\psi\wedge\delta .$$

It is predicative, as a conjunction of predicative formulas, with free variables among $x_{0}$, $x_{1}$ and the parameters. Let sets $P$ and $X$ be assigned to $x_{0}$ and $x_{1}$. If $P$ is not a measure on $(\Omega,\mathcal{F})$ with $(\Omega,\mathcal{F},P)$ a probability space, then $\psi$ fails, and so does $\eta$. If it is one, then $\psi$ holds, so $\eta$ holds if and only if $X\in\mathcal{L}^{1}(P)$, that is, if and only if $X$ is an integrable random variable. Hence $\eta$ is a predicative formula in $x_{0},x_{1}$ expressing the condition of (i) with $P=x_{0}$ and $X=x_{1}$. By Renaming and Substituting Variables in Predicative Formulas (i), with its number of variables equal to $2$, its $n$ equal to $N$ and $\sigma(0)=i$, $\sigma(1)=j$, the formula $\eta(x_{i},x_{j})$ is a predicative formula in $x_{0},\dots,x_{N-1}$ expressing this condition with $P=x_{i}$ and $X=x_{j}$. This proves (i).

The formula for component maps. Let

$$\gamma=\forall u\,\Bigl(u\in x_{4}\Leftrightarrow\exists w\,\exists v\,\exists y\,\bigl(u=(w,v)\wedge w\in\Omega\wedge(w,y)\in x_{2}\wedge(x_{3},v)\in y\bigr)\Bigr).$$

The formulas $u=(w,v)$, $(w,y)\in x_{2}$ and $(x_{3},v)\in y$ are predicative (Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (ii), Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iii)), $u\in x_{4}$ and $w\in\Omega$ are atomic, and every quantifier of $\gamma$ is over a set variable; so $\gamma$ is predicative (Formulas of the Language of Class Theory, Free Variables, Predicative Formulas and Abbreviations (v)), with free variables among $x_{2}$, $x_{3}$, $x_{4}$ and $\Omega$. Let $n\in\mathbb{N}$, let $X:\Omega\to\mathbb{R}^{n}$ be a map, let $c\in[n]$ and let $g$ be a set, assigned to $x_{2}$, $x_{3}$ and $x_{4}$. For $w\in\Omega$ and a set $y$, $(w,y)\in X$ if and only if $y=X(w)$, as $X$ is a function with domain $\Omega$ (Functions, Values of a Function, and Functions from One Class to Another (ii)). The point $X(w)$ is a map $[n]\to\mathbb{R}$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii)), and for a set $v$, $(c,v)\in X(w)$ if and only if

$$v=X(w)(c)=X(w)_{c}=X_{c}(w),$$

the component $X(w)_{c}$ used by Component Maps and Entry Maps being $X(w)(c)$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (ii)) and $X_{c}$ the component map of Component Maps and Entry Maps (i). Hence the right side of the biconditional in $\gamma$ holds for a set $u$ if and only if $u=(w,X_{c}(w))$ for some $w\in\Omega$, that is, if and only if $u\in X_{c}$, the map $X_{c}:\Omega\to\mathbb{R}$ being a class of ordered pairs (Functions, Values of a Function, and Functions from One Class to Another (i)). So $\gamma$ holds if and only if $g$ and $X_{c}$ have the same elements, that is,

$$\gamma\ \text{holds}\quad\Longleftrightarrow\quad g=X_{c};$$

and $X_{c}$ is a set, an element of the set $\mathbb{R}^{\Omega}$ (Component Maps and Entry Maps (i)).

The formula for entry maps. Let

$$\gamma'=\forall u\,\Bigl(u\in x_{6}\Leftrightarrow\exists w\,\exists v\,\exists y\,\bigl(u=(w,v)\wedge w\in\Omega\wedge(w,y)\in x_{3}\wedge((x_{4},x_{5}),v)\in y\bigr)\Bigr).$$

As for $\gamma$, $\gamma'$ is predicative (Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (ii), Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iii), Predicativity under Renaming of Set Variables by Variables, of Formulas with Ordered Pairs, and of Formulas for Relations, Functions, Maps, Injectivity, Surjectivity and Bijectivity (iv)), with free variables among $x_{3}$, $x_{4}$, $x_{5}$, $x_{6}$ and $\Omega$. Let $p,q\in\mathbb{N}$, let $X:\Omega\to\mathbb{R}^{p\times q}$ be a map, let $a\in[p]$, $b\in[q]$, and let $g$ be a set, assigned to $x_{3}$, $x_{4}$, $x_{5}$ and $x_{6}$. For $w\in\Omega$ and a set $y$, $(w,y)\in X$ if and only if $y=X(w)$ (Functions, Values of a Function, and Functions from One Class to Another (ii)). The matrix $X(w)$ is a map $[p]\times[q]\to\mathbb{R}$ (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (i)), and for a set $v$, $((a,b),v)\in X(w)$ if and only if

$$v=X(w)(a,b)=X(w)_{ab}=X_{ab}(w),$$

the entry $X(w)_{ab}$ used by Component Maps and Entry Maps being $X(w)(a,b)$ (Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (ii)) and $X_{ab}$ the entry map of Component Maps and Entry Maps (ii). As for $\gamma$,

$$\gamma'\ \text{holds}\quad\Longleftrightarrow\quad g=X_{ab};$$

and $X_{ab}$ is a set, an element of $\mathbb{R}^{\Omega}$ (Component Maps and Entry Maps (ii)).

The formula for segments. By Induction over Finite Sets, and Predicative Formulas for the Segments [n] and for Finiteness (i), with its $Y$ and $R$ the parameters $\mathbb{N}$ and $R$ of Kit for def:measurable-function, whose values are $\mathbb{N}$ and the order $\le$ of $\mathbb{N}_{0}$, and with its $z$ equal to $z$ and its $n$ equal to $x_{1}$, respectively $x_{2}$, there are predicative formulas $\mathrm{seg}(z,x_{1})$ and $\mathrm{seg}(z,x_{2})$ which, for every set $z$ and every $n\in\mathbb{N}$ assigned to $x_{1}$, respectively $x_{2}$ (an element of $\mathbb{N}_{0}$, The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii)), hold if and only if $z=[n]$. So, for $n\in\mathbb{N}$ assigned to $x_{1}$ and a set $c$ assigned to $x_{3}$, the formula $\exists z\,(\mathrm{seg}(z,x_{1})\wedge x_{3}\in z)$ holds if and only if $c\in[n]$; likewise for $x_{1}$ and $x_{4}$, and for $x_{2}$ and $x_{5}$.

Integrable random vectors. By Kit for def:measurable-function (iii), with its $N$ equal to $4$ and $i=0$, $j=1$, $k=2$, $l=3$, there is a predicative formula $\rho$ in $x_{0},\dots,x_{3}$ which, for sets $Y$, $\mathcal{G}$, $n$ and $f$ assigned to $x_{0},\dots,x_{3}$, holds if and only if $(Y,\mathcal{G})$ is a measurable space, $n\in\mathbb{N}$, and $f:Y\to\mathbb{R}^{n}$ is a map that is measurable as a map into $\mathbb{R}^{n}$ (Measurable Map, and the Set of Measurable Maps (iv)). By Renaming and Substituting Variables in Predicative Formulas (iii), with its $k$ equal to $4$, its $n$ equal to $3$ and $\tau_{0}=\Omega$, $\tau_{1}=\mathcal{F}$, $\tau_{2}=x_{1}$, $\tau_{3}=x_{2}$, the formula $\rho(\Omega,\mathcal{F},x_{1},x_{2})$ is a predicative formula in $x_{0},x_{1},x_{2}$; as $(\Omega,\mathcal{F})$ is a measurable space, for sets $n$ and $X$ assigned to $x_{1}$ and $x_{2}$ it holds if and only if $n\in\mathbb{N}$ and $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ (Measurable Map, and the Set of Measurable Maps (ii)). Let

$$\theta=\psi\wedge\rho(\Omega,\mathcal{F},x_{1},x_{2})\wedge\forall x_{3}\,\Bigl(\exists z\,\bigl(\mathrm{seg}(z,x_{1})\wedge x_{3}\in z\bigr)\Rightarrow\exists x_{4}\,\bigl(\gamma\wedge\delta(x_{0},x_{4})\bigr)\Bigr).$$

Its parts are predicative and its added quantifiers are over set variables, so $\theta$ is predicative, with free variables among $x_{0}$, $x_{1}$, $x_{2}$ and the parameters.

Let sets $P$, $n$ and $X$ be assigned to $x_{0}$, $x_{1}$ and $x_{2}$. Suppose that $\theta$ holds. Then $P$ is a measure on $(\Omega,\mathcal{F})$ with $(\Omega,\mathcal{F},P)$ a probability space; $n\in\mathbb{N}$; and $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, in particular $X$ is a map $X:\Omega\to\mathbb{R}^{n}$. Let $c\in[n]$, a set. By the last conjunct, with $c$ assigned to $x_{3}$ (its premise holds by the segment paragraph), there is a set $g$ such that $\gamma$ and $\delta(x_{0},x_{4})$ hold with $g$ assigned to $x_{4}$; then $g=X_{c}$ and $g\in\mathcal{L}^{1}(P)$. So $X$ satisfies $(*)$, and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$. Conversely, suppose that $P$ is a measure on $(\Omega,\mathcal{F})$ with $(\Omega,\mathcal{F},P)$ a probability space, $n\in\mathbb{N}$ and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$. Then $\psi$ holds, and $\rho(\Omega,\mathcal{F},x_{1},x_{2})$ holds as $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$. If a set $c$ assigned to $x_{3}$ satisfies the premise of the last conjunct, then $c\in[n]$; the set $X_{c}$, assigned to $x_{4}$, satisfies $\gamma$, and it satisfies $\delta(x_{0},x_{4})$ since $X_{c}\in\mathcal{L}^{1}(P)$ by $(*)$. So $\theta$ holds. By the description of integrable random vectors above, $\theta$ is therefore a predicative formula in $x_{0},x_{1},x_{2}$ expressing the condition of (ii) with $P=x_{0}$, $n=x_{1}$ and $X=x_{2}$. By Renaming and Substituting Variables in Predicative Formulas (i), with its number of variables equal to $3$, its $n$ equal to $N$ and $\sigma(0)=i$, $\sigma(1)=j$, $\sigma(2)=k$, the formula $\theta(x_{i},x_{j},x_{k})$ is a predicative formula in $x_{0},\dots,x_{N-1}$ expressing this condition with $P=x_{i}$, $n=x_{j}$ and $X=x_{k}$. This proves (ii).

Integrable matrix-valued functions. By Kit for def:measurable-function (iv), with its $N$ equal to $5$ and $i=0$, $j=1$, $k=2$, $l=3$, $m=4$, there is a predicative formula $\rho'$ in $x_{0},\dots,x_{4}$ which, for sets $Y$, $\mathcal{G}$, $p$, $q$ and $f$ assigned to $x_{0},\dots,x_{4}$, holds if and only if $(Y,\mathcal{G})$ is a measurable space, $p,q\in\mathbb{N}$, and $f:Y\to\mathbb{R}^{p\times q}$ is a map that is measurable as a map into $\mathbb{R}^{p\times q}$ (Measurable Map, and the Set of Measurable Maps (v)). By Renaming and Substituting Variables in Predicative Formulas (iii), with its $k$ equal to $5$, its $n$ equal to $4$ and $\tau_{0}=\Omega$, $\tau_{1}=\mathcal{F}$, $\tau_{2}=x_{1}$, $\tau_{3}=x_{2}$, $\tau_{4}=x_{3}$, the formula $\rho'(\Omega,\mathcal{F},x_{1},x_{2},x_{3})$ is a predicative formula in $x_{0},\dots,x_{3}$; as $(\Omega,\mathcal{F})$ is a measurable space, for sets $p$, $q$ and $X$ assigned to $x_{1}$, $x_{2}$ and $x_{3}$ it holds if and only if $p,q\in\mathbb{N}$ and $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$ (Measurable Map, and the Set of Measurable Maps (ii)). Let

$$\begin{aligned} \theta'={}&\psi\wedge\rho'(\Omega,\mathcal{F},x_{1},x_{2},x_{3})\\ &\wedge\forall x_{4}\,\forall x_{5}\,\Bigl(\exists z\,\bigl(\mathrm{seg}(z,x_{1})\wedge x_{4}\in z\bigr)\wedge\exists z\,\bigl(\mathrm{seg}(z,x_{2})\wedge x_{5}\in z\bigr)\Rightarrow\exists x_{6}\,\bigl(\gamma'\wedge\delta(x_{0},x_{6})\bigr)\Bigr). \end{aligned}$$

Its parts are predicative and its added quantifiers are over set variables, so $\theta'$ is predicative, with free variables among $x_{0},\dots,x_{3}$ and the parameters.

Let sets $P$, $p$, $q$ and $X$ be assigned to $x_{0},\dots,x_{3}$. Suppose that $\theta'$ holds. Then $P$ is a measure on $(\Omega,\mathcal{F})$ with $(\Omega,\mathcal{F},P)$ a probability space; $p,q\in\mathbb{N}$; and $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$, in particular $X$ is a map $X:\Omega\to\mathbb{R}^{p\times q}$. Let $a\in[p]$ and $b\in[q]$. By the last conjunct, with $a$ and $b$ assigned to $x_{4}$ and $x_{5}$ (its premise holds by the segment paragraph), there is a set $g$ such that $\gamma'$ and $\delta(x_{0},x_{6})$ hold with $g$ assigned to $x_{6}$; then $g=X_{ab}$ and $g\in\mathcal{L}^{1}(P)$. So $X$ satisfies $(**)$, and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$. Conversely, suppose that $P$ is a measure on $(\Omega,\mathcal{F})$ with $(\Omega,\mathcal{F},P)$ a probability space, $p,q\in\mathbb{N}$ and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{p\times q})$. Then $\psi$ holds, and $\rho'(\Omega,\mathcal{F},x_{1},x_{2},x_{3})$ holds as $X\in\mathrm{Meas}(\Omega,\mathcal{F};\mathbb{R}^{p\times q},\mathcal{B}(\mathbb{R}^{p\times q}))$. If sets $a$ and $b$ assigned to $x_{4}$ and $x_{5}$ satisfy the premise of the last conjunct, then $a\in[p]$ and $b\in[q]$; the set $X_{ab}$, assigned to $x_{6}$, satisfies $\gamma'$, and it satisfies $\delta(x_{0},x_{6})$ since $X_{ab}\in\mathcal{L}^{1}(P)$ by $(**)$. So $\theta'$ holds. Hence $\theta'$ is a predicative formula in $x_{0},\dots,x_{3}$ expressing the condition of (iii) with $P=x_{0}$, $p=x_{1}$, $q=x_{2}$ and $X=x_{3}$. By Renaming and Substituting Variables in Predicative Formulas (i), with its number of variables equal to $4$, its $n$ equal to $N$ and $\sigma(0)=i$, $\sigma(1)=j$, $\sigma(2)=k$, $\sigma(3)=l$, the formula $\theta'(x_{i},x_{j},x_{k},x_{l})$ is a predicative formula in $x_{0},\dots,x_{N-1}$ expressing this condition with $P=x_{i}$, $p=x_{j}$, $q=x_{k}$ and $X=x_{l}$. This proves (iii).

The classes. Let $P$ be a measure on $(\Omega,\mathcal{F})$ such that $(\Omega,\mathcal{F},P)$ is a probability space, and let $n\in\mathbb{N}$. As shown above, $(\Omega,\mathcal{F},P)$ is a measure space, so the sets $\mathcal{L}^{1}(P)$ of Integrable Functions and Their Lebesgue Integral (ii) and $\mathcal{L}^{1}(P;\mathbb{R}^{n})$ of Integrable Vector- and Matrix-Valued Functions and Their Integrals (i) are defined.

For (iv), let $C=\mathcal{L}^{1}(P)$, a set. For every class $a$, $a\in C$ if and only if $a\in\mathcal{L}^{1}(P)$, which holds if and only if $a$ is an integrable random variable, as shown above. This proves (iv).

For (v), let $D=\mathcal{L}^{1}(P;\mathbb{R}^{n})$, a set. For every class $a$, $a\in D$ if and only if $a\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$, which holds if and only if $a$ is an integrable random vector in $\mathbb{R}^{n}$, as shown above. This proves (v).