= Doob-Dynkin lemma
{c}
{title2=$Z\text{ is }\sigma(X)\text{-measurable}\Longrightarrow Z=g(X)$}
For real <random variables> $X,Z$, if $Z$ is <measurable> with respect to the <sigma-algebra generated by a random variable> $X$, there is a <Borel measurable function> $g:\mathbb R\to\mathbb R$ with $Z=g(X)$. To prove the bounded case, approximate $Z$ pointwise by finite-valued <simple functions> $Z_m$. Each level <set> is $X^{-1}(B)$ for a <Borel set> $B$, so each approximation factors as $Z_m=g_m(X)$ with $g_m$ a <Borel measurable function>. Apply <scalar clipping> to $g_m$ with a common bounded interval and use its <limit superior> to obtain $g$. For unbounded finite $Z$, apply the bounded case to the one-to-one bounded <continuous function> $t\mapsto\arctan t$, then invert on $(-\pi/2,\pi/2)$ and assign arbitrary values elsewhere. Exact <measurability> gives exact factorization; assumptions after adjoining <null sets> give an <almost surely> equal version.
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