Proof of Existence and Uniqueness of Conditional Expectations

Throughout, $P|_{\mathcal{G}}$ denotes the restriction of $P$ to $\mathcal{G}$, a probability measure on $(\Omega,\mathcal{G})$ (Finite Signed Measures from Finite Measures and from Integrals (iii)); it is $\sigma$-finite, the constant sequence $\Omega$ witnessing this (Measure, Measure Space, and Probability Measure (iv)), so the Radon–Nikodym theorem applies in the measure space $(\Omega,\mathcal{G},P|_{\mathcal{G}})$. A $\mathcal{G}$-measurable $Y:\Omega\to\mathbb{R}$ is $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable, it lies in $\mathcal{L}^{1}(\Omega,\mathcal{G},P|_{\mathcal{G}})$ if and only if it lies in $\mathcal{L}^{1}(P)$, and then

$$\int_{G}Y\,d(P|_{\mathcal{G}})=\int_{G}Y\,dP\qquad(G\in\mathcal{G})$$

(Integrals with Respect to the Restriction of a Measure to a Sub-σ-Algebra (ii), Integrals with Respect to the Restriction of a Measure to a Sub-σ-Algebra (iv), with Integrals over Measurable Subsets (vi) for integrability on $G$).

Existence. Let $X\in\mathcal{L}^{1}(P)$. The function $\nu$ on $\mathcal{G}$,

$$\nu(G)=\int_{G}X\,dP\qquad(G\in\mathcal{G}),$$

is a finite signed measure on $(\Omega,\mathcal{G})$ that is absolutely continuous with respect to $P|_{\mathcal{G}}$ (Finite Signed Measures from Finite Measures and from Integrals (iv)). Under the axiom of dependent choice, the Radon–Nikodym theorem for finite signed measures, applied in $(\Omega,\mathcal{G},P|_{\mathcal{G}})$ (The Radon–Nikodym Theorem for a σ-Finite Measure, and Uniqueness of Densities (iv)), gives $Y\in\mathcal{L}^{1}(\Omega,\mathcal{G},P|_{\mathcal{G}})$ with $\nu(G)=\int_{G}Y\,d(P|_{\mathcal{G}})$ for every $G\in\mathcal{G}$. As an element of $\mathcal{L}^{1}(\Omega,\mathcal{G},P|_{\mathcal{G}})$, $Y$ is $\mathcal{G}$-measurable (Integrable Functions and Their Lebesgue Integral (ii)), so by the transfer above $Y\in\mathcal{L}^{1}(P)$ and

$$\int_{G}Y\,dP=\int_{G}Y\,d(P|_{\mathcal{G}})=\nu(G)=\int_{G}X\,dP\qquad(G\in\mathcal{G}).$$

This proves (i).

Uniqueness. Let $Y$ and $Y'$ be as in (ii). By the transfer, both lie in $\mathcal{L}^{1}(\Omega,\mathcal{G},P|_{\mathcal{G}})$ and

$$\int_{G}Y\,d(P|_{\mathcal{G}})=\int_{G}X\,dP=\int_{G}Y'\,d(P|_{\mathcal{G}})\qquad(G\in\mathcal{G}).$$

The uniqueness of densities in $(\Omega,\mathcal{G},P|_{\mathcal{G}})$ (The Radon–Nikodym Theorem for a σ-Finite Measure, and Uniqueness of Densities (i)), which uses no choice, shows that $\{Y\neq Y'\}$ belongs to $\mathcal{G}$ and has $P|_{\mathcal{G}}$-measure $0$, that is, $P(Y\neq Y')=0$.

The vector case. Let $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$; its components $X_{k}$ lie in $\mathcal{L}^{1}(P)$ (Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)). A function $Y:\Omega\to\mathbb{R}^{n}$ is $\mathcal{G}$-measurable, respectively $\mathcal{F}/\mathcal{B}(\mathbb{R}^{n})$-measurable, if and only if all its components are $\mathcal{G}$-measurable, respectively $\mathcal{F}/\mathcal{B}(\mathbb{R})$-measurable (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), applied on $(\Omega,\mathcal{G})$ and on $(\Omega,\mathcal{F})$).

For uniqueness, let $Y$ and $Y'$ be as in (iii). For each $k\in[n]$, (ii) applied to $X_{k}$, $Y_{k}$ and $Y'_{k}$ shows that $N_{k}=\{Y_{k}\neq Y'_{k}\}$ lies in $\mathcal{G}$ with $P(N_{k})=0$. Since $\{Y\neq Y'\}=\bigcup_{k\in[n]}N_{k}$, by induction on $n$ this set lies in $\mathcal{G}$ and $P(Y\neq Y')\le\sum_{k=1}^{n}P(N_{k})=0$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vii), Basic Properties of a Measure (vii)). This proves (iii); no choice is used.

For existence, assume the axiom of dependent choice. For $k\in[n]$ let $\mathcal{V}_{k}$ be the set of $\mathcal{G}$-measurable $W\in\mathcal{L}^{1}(P)$ with $\int_{G}W\,dP=\int_{G}X_{k}\,dP$ for every $G\in\mathcal{G}$; it is nonempty by (i). Since $[n]$ is finite (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vii)), finite choice (Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (xi)), applied to the family $(\mathcal{V}_{k})_{k\in[n]}$ of nonempty sets, gives a family $(W_{k})_{k\in[n]}$ with $W_{k}\in\mathcal{V}_{k}$ for every $k\in[n]$. Let $Y:\Omega\to\mathbb{R}^{n}$ be the function with components $Y_{k}=W_{k}$ for $k\in[n]$ (Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii)). Its components are $\mathcal{G}$-measurable and lie in $\mathcal{L}^{1}(P)$, so $Y$ is $\mathcal{G}$-measurable, $\mathcal{F}/\mathcal{B}(\mathbb{R}^{n})$-measurable, and $Y\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$, with $\int_{G}Y_{k}\,dP=\int_{G}X_{k}\,dP$ for every $G\in\mathcal{G}$ and $k\in[n]$. This proves (iv), using no choice beyond (i).

Uses of dependent choice. The existence statements use the axiom of dependent choice only through The Radon–Nikodym Theorem for a σ-Finite Measure, and Uniqueness of Densities (iv); the uniqueness statements use no choice.