Proof of The Doob–Dynkin Factorization

Write $\mathcal{C}_{Z}=\{Z^{-1}(B):B\in\mathcal{B}(S)\}$, so that $\sigma(Z)=\sigma(\mathcal{C}_{Z})$ is a $\sigma$-algebra on $\Omega$ contained in $\mathcal{F}$ (σ-Algebras Generated by Random Elements (i)), and let $\mathcal{H}$ be the set of functions $\Omega\to\mathbb{R}$ of the form $h(Z)$ with $h:S\to\mathbb{R}$ Borel.

Step 1: sets of $\sigma(Z)$ (countable choice). We show that $\mathcal{C}_{Z}$ is a $\sigma$-algebra on $\Omega$. It contains $\Omega=Z^{-1}(S)$, and for $B\in\mathcal{B}(S)$ the complement $\Omega\setminus Z^{-1}(B)=Z^{-1}(S\setminus B)$ lies in $\mathcal{C}_{Z}$, as $\mathcal{B}(S)$ is a $\sigma$-algebra on $S$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii)). Let $(A_{m})$ be a sequence in $\mathcal{C}_{Z}$. For each $m$ the set of $B\in\mathcal{B}(S)$ with $A_{m}=Z^{-1}(B)$ is nonempty; under the axiom of dependent choice, countable choice (Countable Choice Follows from Dependent Choice (i)) gives a sequence $(B_{m})$ in $\mathcal{B}(S)$ with $A_{m}=Z^{-1}(B_{m})$ for every $m$, and then

$$\bigcup_{m\in\mathbb{N}}A_{m}=Z^{-1}\Bigl(\bigcup_{m\in\mathbb{N}}B_{m}\Bigr)\in\mathcal{C}_{Z}$$

(Preimages of Unions, Intersections and Complements). So $\mathcal{C}_{Z}$ is a $\sigma$-algebra (σ-Algebra and Measurable Space (i)), and $\sigma(Z)\subseteq\mathcal{C}_{Z}$ (Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iv)): every $A\in\sigma(Z)$ is $Z^{-1}(B)$ for some $B\in\mathcal{B}(S)$. For such $A$ and $B$, $\mathbf{1}_{A}=\mathbf{1}_{B}(Z)$, and $\mathbf{1}_{B}$ is Borel (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (v): its strict upper level sets are $\emptyset$, $B$ or $S$). Hence $\mathbf{1}_{A}\in\mathcal{H}$ for every $A\in\sigma(Z)$.

Step 2: simple functions. If $h_{1}(Z),h_{2}(Z)\in\mathcal{H}$ and $c\in\mathbb{R}$, then $h_{1}(Z)+h_{2}(Z)=(h_{1}+h_{2})(Z)$ and $c\,h_{1}(Z)=(ch_{1})(Z)$ lie in $\mathcal{H}$, as $h_{1}+h_{2}$ and $ch_{1}$ are Borel (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i)). Let $t$ be a simple function on $(\Omega,\sigma(Z))$. Its image $t[\Omega]$ is finite, each level set $t^{-1}(\{c\})$, $c\in t[\Omega]$, lies in $\sigma(Z)$ because $t$ is $\sigma(Z)/\mathcal{B}(\mathbb{R})$-measurable and $\{c\}$ is a Borel set (Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (iii)), and

$$t=\sum_{c\in t[\Omega]}c\,\mathbf{1}_{t^{-1}(\{c\})},$$

since each $\omega$ lies in exactly one of these level sets, that of $c=t(\omega)$. For subsets $F$ of the finite set $t[\Omega]$, induction on the number of elements of $F$ shows that $\sum_{c\in F}c\,\mathbf{1}_{t^{-1}(\{c\})}\in\mathcal{H}$: for $F=\emptyset$ the sum is the zero function $\mathbf{1}_{\emptyset}$, which lies in $\mathcal{H}$ by Step 1, and adding one term keeps the sum in $\mathcal{H}$ by Step 1 and the closure under sums and multiples. Taking $F=t[\Omega]$ gives $t\in\mathcal{H}$.

Step 3: limits (countable choice). Let $Y$ be as in (i). By Approximation of Measurable Functions by Dyadic Simple Functions (iii), applied on the measurable space $(\Omega,\sigma(Z))$, there are simple functions $t_{m}$ on $(\Omega,\sigma(Z))$ with $t_{m}(\omega)\to Y(\omega)$ for every $\omega\in\Omega$. By Step 2 the set of Borel $h:S\to\mathbb{R}$ with $t_{m}=h(Z)$ is nonempty for each $m$; countable choice (Countable Choice Follows from Dependent Choice (i)) gives a sequence $(h_{m})$ of Borel functions with $t_{m}=h_{m}(Z)$ for every $m$. Let

$$C=\{s\in S:(h_{m}(s))_{m\in\mathbb{N}}\text{ converges}\},$$

a Borel set (Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps (iii)). The functions $h_{m}\mathbf{1}_{C}$ are Borel (Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), and Step 1 for $\mathbf{1}_{C}$), and $(h_{m}(s)\mathbf{1}_{C}(s))_{m}$ converges for every $s\in S$, to $\lim_{m}h_{m}(s)$ on $C$ and to $0$ off $C$. So

$$h(s)=\lim_{m\to\infty}h_{m}(s)\mathbf{1}_{C}(s)\qquad(s\in S)$$

defines a Borel function $h:S\to\mathbb{R}$ (Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps (ii)). For $\omega\in\Omega$ the sequence $(h_{m}(Z(\omega)))_{m}=(t_{m}(\omega))_{m}$ converges to $Y(\omega)$, so $Z(\omega)\in C$ and $h(Z(\omega))=Y(\omega)$. Thus $Y=h(Z)$, which proves (i).

Factoring a version. Let $X\in\mathcal{L}^{1}(P)$. Under the axiom of dependent choice, Existence and Uniqueness of Conditional Expectations (i) with $\mathcal{G}=\sigma(Z)$ gives a $\sigma(Z)$-measurable $Y\in\mathcal{L}^{1}(P)$ with $\int_{G}Y\,dP=\int_{G}X\,dP$ for every $G\in\sigma(Z)$, that is, $Y\in\mathbb{E}[X\mid\sigma(Z)]$ (Conditional Expectation (ii)). By (i) there is a Borel $h:S\to\mathbb{R}$ with $h(Z)=Y\in\mathbb{E}[X\mid\sigma(Z)]$. This proves (ii).

Now let $n\in\mathbb{N}$ and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$. By Existence and Uniqueness of Conditional Expectations (iv) there is a $\sigma(Z)$-measurable $Y\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$ with $\int_{G}Y_{k}\,dP=\int_{G}X_{k}\,dP$ for every $G\in\sigma(Z)$ and $k\in[n]$; its components are $\sigma(Z)$-measurable and lie in $\mathcal{L}^{1}(P)$ (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (i)), so $Y_{k}\in\mathbb{E}[X_{k}\mid\sigma(Z)]$ (Conditional Expectation (ii)) and $Y\in\mathbb{E}[X\mid\sigma(Z)]$ (Conditional Expectation (iii)). For $k\in[n]$ the set $\mathcal{D}_{k}$ of Borel $g:S\to\mathbb{R}$ with $Y_{k}=g(Z)$ is nonempty by (i). By induction on $m\in\{0,\dots,n\}$ there is a family $(g_{k})_{k\in[m]}$ with $g_{k}\in\mathcal{D}_{k}$ for every $k\in[m]$: for $m=0$ the empty family, and for $m<n$ a family for $m$ extended by one element of the nonempty set $\mathcal{D}_{m+1}$. This finite choice needs no axiom. Taking $m=n$, let $h:S\to\mathbb{R}^{n}$ be the function with components $h_{k}=g_{k}$ (Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (iii)); it is Borel (Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images (i), on $(S,\mathcal{B}(S))$), and $h(Z)=Y\in\mathbb{E}[X\mid\sigma(Z)]$. This proves (iii).

Uses of dependent choice. Countable choice is used in Steps 1 and 3, and dependent choice once more through Existence and Uniqueness of Conditional Expectations (i) and Existence and Uniqueness of Conditional Expectations (iv); the vector case adds only finite choice.