Let $(X,d)$ be a separable metric space with Borel measure $\mu$. Let $f:X \times X \to \mathbb{R}$ be Borel measurable with respect to the product measure on $X \times X$, and let $g(x)=\operatorname{ess sup}\_{y \in X} f(x,y)$. Is $g(x)$ necessarily measurable? (Is there some argument that can be pieced together using separability of $X$ and Lusin's Theorem, if we assume that $\mu$ is a Radon measure?)



Measurability of essential supremum of function of two variables
You are right. For each $n$ choose a set of measure less than $1/n$ on the complement of which $f$ is continuous. Now take the actual sup on each vertical section of this restricted function. This yields a measurable function $f\_n$ for each $n$ defined on $X$. The sup of the increasing sequence of $f\_n$ will also be a measurable function $F$. Except for a null set, $F$ will give the $\operatorname{esssup}$ of the vertical section of $f$. So modifying $F$ on a null set yields that $g$ is measurable.


