Proof of Rules for Conditional Expectations

Throughout, the field and order laws of $\mathbb{R}$ are used freely (The Structure of ℝ as a Field and an Ordered Ring (i), The Structure of ℝ as a Field and an Ordered Ring (ii), Rules of Arithmetic and Order in an Ordered Field (ii), Rules of Arithmetic and Order in an Ordered Field (iv), Rules of Arithmetic and Order in an Ordered Field (v), Rules of Arithmetic and Order in an Ordered Field (vi), Rules of Arithmetic and Order in an Ordered Field (x)). A function $Z$ belongs to $\mathbb{E}[X\mid\mathcal{G}]$ if and only if $Z\in\mathcal{L}^{1}(P)$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable with $\int_{G}Z\,dP=\int_{G}X\,dP$ for every $G\in\mathcal{G}$ (Conditional Expectation (ii); $\mathcal{G}$-measurability of real functions is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurability by Conditional Expectations: Notation (ii)). For $f\in\mathcal{L}^{1}(P)$ and $A\in\mathcal{F}$, $f$ is integrable on $A$ and $\int_{A}f\,dP=\mathbb{E}[f\mathbf{1}_{A}]$ (Integrals over Measurable Subsets (vi), Integrals over Measurable Subsets (ii)). A $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable function is $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable, as $\mathcal{G}\subseteq\mathcal{F}$ (Measurable Map, and the Set of Measurable Maps (i)); sums, differences, products, real multiples and absolute values of $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable functions are $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable, and so are constants and the indicators $\mathbf{1}_{G}$ of sets $G\in\mathcal{G}$ (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii) and Properties of Simple Functions and of the Simple Integral (i), all for the measurable space $(\Omega,\mathcal{G})$); and for $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable $S,T$ the set $\{S>T\}=\{S-T>0\}$ lies in $\mathcal{G}$ (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v)).

Proof of (i). $X$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable and integrable, and $\int_{G}X\,dP=\int_{G}X\,dP$ for every $G\in\mathcal{G}$; so $X$ is a version of $\mathbb{E}[X\mid\mathcal{G}]$.

Proof of (ii). $\Omega\in\mathcal{G}$, so $\int_{\Omega}Y\,dP=\int_{\Omega}X\,dP$, and these integrals are $\mathbb{E}Y$ and $\mathbb{E}X$ (Integrals over Measurable Subsets (v)).

Proof of (iii). The function $Z=aY+bY'$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable, and $Z$ and $aX+bX'$ are integrable (Rules for Expectations, Variances and Covariances (iii)). For $G\in\mathcal{G}$ we have $Z\mathbf{1}_{G}=a\,Y\mathbf{1}_{G}+b\,Y'\mathbf{1}_{G}$, so by linearity of the expectation (Rules for Expectations, Variances and Covariances (iii)) and the defining identities of $Y$ and $Y'$,

$$\int_{G}Z\,dP=a\,\mathbb{E}[Y\mathbf{1}_{G}]+b\,\mathbb{E}[Y'\mathbf{1}_{G}]=a\int_{G}X\,dP+b\int_{G}X'\,dP=\mathbb{E}\bigl[(aX+bX')\mathbf{1}_{G}\bigr]=\int_{G}(aX+bX')\,dP .$$

Proof of (iv). Let $G=\{Y>Y'\}\in\mathcal{G}$ and $D=Y-Y'$, a version of $\mathbb{E}[X-X'\mid\mathcal{G}]$ by (iii). The function $f=(X-X')\mathbf{1}_{G}$ is integrable, and the measurable function $g=\min(f,0)$ (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)) agrees with $f$ at every $\omega$ with $X(\omega)\le X'(\omega)$, hence almost surely (Null Sets and Properties Holding Almost Everywhere (vi)). So $g$ is integrable with $\mathbb{E}f=\mathbb{E}g$ (The Lebesgue Integral and Null Sets (iii)), and $\mathbb{E}g\le0$ (Rules for Expectations, Variances and Covariances (iv), Rules for Expectations, Variances and Covariances (ii)). Hence

$$\mathbb{E}[D\mathbf{1}_{G}]=\int_{G}D\,dP=\int_{G}(X-X')\,dP=\mathbb{E}f\le0 .$$

On the other hand $D\mathbf{1}_{G}\ge0$, so its integral as an element of $\mathcal{M}^{+}$ is $\mathbb{E}[D\mathbf{1}_{G}]$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (vi)), which is therefore $0$. Since $\{D\mathbf{1}_{G}>0\}=G$, The Lebesgue Integral and Null Sets (iv) gives $P(G)=0$: the set $\{Y>Y'\}$ is null (Null Sets and Properties Holding Almost Everywhere (i)), that is, $Y\le Y'$ a.s.

Proof of (v). Let $V\in\mathbb{E}[|X|\mid\mathcal{G}]$, where $|X|\in\mathcal{L}^{1}(P)$ (Rules for Expectations, Variances and Covariances (v)). By (iii) (with $b=0$), $-V\in\mathbb{E}[-|X|\mid\mathcal{G}]$. As $-|X|\le X\le|X|$, (iv) gives $-V\le Y$ a.s. and $Y\le V$ a.s., so both hold a.s. (Null Sets and Properties Holding Almost Everywhere (vii)), and at every such $\omega$ we have $|Y(\omega)|\le V(\omega)$; hence $|Y|\le V$ a.s. (Null Sets and Properties Holding Almost Everywhere (vi)).

Proof of (vi). Both sides are defined, as $X,Y\in\mathcal{L}^{1}(P)$ and $\mathcal{H}\subseteq\mathcal{F}$. A function $Z$ belongs to $\mathbb{E}[Y\mid\mathcal{H}]$ if and only if it is an $\mathcal{H}/\mathcal{B}(\mathbb{R})$-measurable element of $\mathcal{L}^{1}(P)$ with $\int_{H}Z\,dP=\int_{H}Y\,dP$ for every $H\in\mathcal{H}$, and to $\mathbb{E}[X\mid\mathcal{H}]$ if and only if the same holds with $X$ in place of $Y$. Every $H\in\mathcal{H}$ lies in $\mathcal{G}$, so $\int_{H}Y\,dP=\int_{H}X\,dP$, and the two conditions coincide.

Proof of (vii). The function $WY$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable and integrable, so it remains to show, for fixed $G\in\mathcal{G}$,

$$\mathbb{E}[WY\mathbf{1}_{G}]=\mathbb{E}[WX\mathbf{1}_{G}] .$$

Indicators. If $W=\mathbf{1}_{G'}$ with $G'\in\mathcal{G}$, then $WY\mathbf{1}_{G}=Y\mathbf{1}_{G\cap G'}$ and $WX\mathbf{1}_{G}=X\mathbf{1}_{G\cap G'}$ with $G\cap G'\in\mathcal{G}$, and the identity is the defining identity of $Y$ on $G\cap G'$.

Simple functions. Let $W$ be a simple function on $(\Omega,\mathcal{G})$. Its image $W[\Omega]$ is finite, $\{W=c\}\in\mathcal{G}$ for $c\in W[\Omega]$ (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (iii)), and

$$W=\sum_{c\in W[\Omega]}c\,\mathbf{1}_{\{W=c\}} .$$

$W$ is bounded, so all the products below are integrable (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v)), and linearity over this finite sum (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (viii), after listing $W[\Omega]$ as $c_{1},\dots,c_{N}$) and the case of indicators give

$$\mathbb{E}[WY\mathbf{1}_{G}]=\sum_{c\in W[\Omega]}c\,\mathbb{E}\bigl[\mathbf{1}_{\{W=c\}}Y\mathbf{1}_{G}\bigr]=\sum_{c\in W[\Omega]}c\,\mathbb{E}\bigl[\mathbf{1}_{\{W=c\}}X\mathbf{1}_{G}\bigr]=\mathbb{E}[WX\mathbf{1}_{G}] .$$

General $W$. By Approximation of Measurable Functions by Dyadic Simple Functions (iii), applied on $(\Omega,\mathcal{G})$, there are simple functions $t_{m}$ on $(\Omega,\mathcal{G})$ with $|t_{m}|\le|W|$ and $t_{m}(\omega)\to W(\omega)$ for every $\omega$. Then $t_{m}X\mathbf{1}_{G}\to WX\mathbf{1}_{G}$ pointwise (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (vi)) and $|t_{m}X\mathbf{1}_{G}|\le|WX|\in\mathcal{L}^{1}(P)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (x)), so by dominated convergence (Dominated Convergence Theorem (iii)) $\mathbb{E}[t_{m}X\mathbf{1}_{G}]\to\mathbb{E}[WX\mathbf{1}_{G}]$; in the same way $\mathbb{E}[t_{m}Y\mathbf{1}_{G}]\to\mathbb{E}[WY\mathbf{1}_{G}]$. By the case of simple functions the two sequences coincide, so their limits agree (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (ii)).

Proof of (viii). We have $|WX|\le c|X|$ and $|WY|\le c|Y|$ with $c|X|,c|Y|\in\mathcal{L}^{1}(P)$ (Rules for Expectations, Variances and Covariances (v), Rules for Expectations, Variances and Covariances (iii)), so $WX,WY\in\mathcal{L}^{1}(P)$ (Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion (v)), and (vii) applies.

Proof of (ix). The constant $\mathbb{E}X$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable and integrable (Rules for Expectations, Variances and Covariances (ii)). Let $G\in\mathcal{G}$. The random variable $\mathbf{1}_{G}$ is $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable, so $\sigma(\mathbf{1}_{G})\subseteq\mathcal{G}$ (σ-Algebras Generated by Random Elements (i), The σ-Algebra Generated by Measurable Functions into Metric Spaces (i)). As $\sigma(X)$ and $\mathcal{G}$ are independent, so are $\sigma(X)$ and $\sigma(\mathbf{1}_{G})$, directly from Independence of Classes of Events and of σ-Algebras (iii); that is, $X$ and $\mathbf{1}_{G}$ are independent. By the product rule (Expectations and Variances of Independent Random Variables (i)) and Rules for Expectations, Variances and Covariances (i),

$$\int_{G}X\,dP=\mathbb{E}[X\mathbf{1}_{G}]=\mathbb{E}X\cdot P(G)=\mathbb{E}\bigl[(\mathbb{E}X)\mathbf{1}_{G}\bigr]=\int_{G}\mathbb{E}X\,dP,$$

the third equality by Rules for Expectations, Variances and Covariances (iii).

Random vectors. A function $\Omega\to\mathbb{R}^{m}$ whose components are $\mathcal{G}$-measurable elements of $\mathcal{L}^{1}(P)$ is a $\mathcal{G}$-measurable element of $\mathcal{L}^{1}(P;\mathbb{R}^{m})$ (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i) on $(\Omega,\mathcal{G})$ and on $(\Omega,\mathcal{F})$, Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)). So, by Conditional Expectation (iii), a relation $S\in\mathbb{E}[Z\mid\mathcal{G}]$ in one of the last three clauses holds once $Z$ is integrable and $S_{k}\in\mathbb{E}[Z_{k}\mid\mathcal{G}]$ for every index $k$; moreover $V_{k}\in\mathbb{E}[U_{k}\mid\mathcal{G}]$ for $k\in[n]$, where $U_{k}\in\mathcal{L}^{1}(P)$ (Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)), and likewise for $U'$ and $V'$ in (x).

Proof of (x). The random vector $aU+a'U'$ is integrable (Rules for Expectations, Variances and Covariances (iii), applied to the components). For $k\in[n]$ the $k$-th components of $aV+a'V'$ and $aU+a'U'$ are $aV_{k}+a'V'_{k}$ and $aU_{k}+a'U'_{k}$, so the assertion is (iii).

Proof of (xi). The random vector $AU+b$ is integrable (Rules for Expectations, Variances and Covariances (vi)). For $i\in[m]$ the $i$-th components of $AV+b$ and $AU+b$ are $\sum_{j=1}^{n}A_{ij}V_{j}+b_{i}$ and $\sum_{j=1}^{n}A_{ij}U_{j}+b_{i}$ (Matrix-Vector Product (i)); by (iii) and induction on the number of summands, $\sum_{j=1}^{n}A_{ij}V_{j}\in\mathbb{E}\bigl[\sum_{j=1}^{n}A_{ij}U_{j}\bigm|\mathcal{G}\bigr]$; the constant $b_{i}$ belongs to $\mathbb{E}[b_{i}\mid\mathcal{G}]$ by (i), and (iii) once more gives the assertion.

Proof of (xii). For $k\in[n]$, the components $WU_{k}$ and $WV_{k}$ of the integrable random vectors $WU$ and $WV$ are integrable, so $WV_{k}\in\mathbb{E}[WU_{k}\mid\mathcal{G}]$ by (vii), which is the assertion.