Let $(\Omega ,\mathcal{A}, P)$ be a probability space, $\mathcal{F} \subseteq\mathcal{A}$ a sigma algebra and $X\in \mathcal{L}^1(\Omega ,\mathcal{A}, P)$. For $\varepsilon>0$ define $f_\varepsilon :\Omega\rightarrow [0.1]$,
$f_\varepsilon (\omega):=\displaystyle\sup_{a\in\mathbb{R}} P(a<X\leq a+\varepsilon |\mathcal{F})(\omega)$
for a regular version of conditional distribution (RCP).
My Question: Is $f_\varepsilon$ measurable?
My Problem: In the definition of $f_\varepsilon$ we have uncountable random variables $X$ for the supremum...
Can someone show me the measurability or a counterexample?
Tanks for any help!