The Doob–Dynkin Factorization

In the setting of Conditional Expectations: Notation, let $(S,d)$ be a metric space, let $Z$ be a random element of $S$, and let $\sigma(Z)$ be the $\sigma$-algebra generated by $Z$.

(i) Factorization. Assume the axiom of dependent choice. Let $Y:\Omega\to\mathbb{R}$ be $\sigma(Z)$-measurable. Then there is a Borel function $h:S\to\mathbb{R}$ with

$$Y=h(Z).$$

(ii) Conditional expectations given $\sigma(Z)$. Assume the axiom of dependent choice, and let $X\in\mathcal{L}^{1}(P)$. Then there is a Borel function $h:S\to\mathbb{R}$ with $h(Z)$ in the conditional expectation $\mathbb{E}[X\mid\sigma(Z)]$.

(iii) Conditional expectations of random vectors given $\sigma(Z)$. Assume the axiom of dependent choice, and let $n\in\mathbb{N}$ and $X\in\mathcal{L}^{1}(P;\mathbb{R}^{n})$. Then there is a Borel function $h:S\to\mathbb{R}^{n}$ with $h(Z)$ in the conditional expectation $\mathbb{E}[X\mid\sigma(Z)]$.