Let $X,Y$ be measurable spaces and $F\subseteq X\times Y$. We say that $f:X\to Y$ is a uniformization map for $F$ if $(x,f(x))\in F$ for each $x\in \pi_X(F)$ where $\pi_X$ is the left projection map. Alternatively, in the stochastic games/control literature the term "selection map" is used for $f$. A uniformization theorem puts condition on $X,Y$ and $F$ that ensure the existence of $f$ with nice measurability properties.
For example, in the important case when $X$ and $Y$ are standard Borel spaces, the Jankov-von Neumann theorem states that every analytic set $F$ admits an analytically measurable uniformization map (that is $f^{-1}(B)$ belongs to a $\sigma$-algebra generated by analytic subset of $X$ for each Borel $B\subseteq Y$).
Another example is Mackey's result, which states that for standard Borel spaces $X,Y$ and a probability measure $m$ on $X$, each Borel set $F$ satisfying $m(\pi_X(F)) = 1$ admits a $m$-a.e. Borel-measurable selector, so that there exists a map $f$ such that $m(x:(x,f(x))\in F) = 1$.
"Survey of measurable selection theorems" by Wagner (1977) gives a comprehensive list of the measurable selection results known by that time. Was there any significant progress in this topic after this paper was written? For example, some results in Section 18 of "Classical Descriptive Set Theory" does not seem to be included in the survey by Wagner. Maybe there are even more recent surveys on uniformization/measurable selection theorems?
