Let $(\Omega, \mathcal{F})$ be a measurable space and $X$ some metric space (probably Polish) with the Borel $\sigma$-algebra and a function $f: \Omega \times X \to \mathbb{R}$. Usually, functions like this are assumed to be measurable in the first variable and continuous in the second, i.e. for all $x\in X$, $\omega \mapsto f(\omega,x)$ is measurable and for all $\omega \in \Omega$, $x \mapsto f(\omega,x)$ is continuous.
I'm looking for ways to extend this function when it is not defined in all $\Omega \times X$. Let $D\subset X$ a compact (or measurable) subset, if $f:\Omega \times D \to \mathbb{R}$ is measurable in the first variable and for all $\omega \in \Omega$, $x\in D \mapsto f(\omega, x)$ is continuous it's easy to extend the function so that $\tilde{f}:\Omega\times X\to\mathbb{R}$ is continuous in all $X$ and $\tilde{f}|_D=f$.
But what I'm looking for is some way to extend when the function is defined on a subset $F\subset \Omega \times X$, with $Pr_{\Omega}(F)\neq \Omega$, where $f$ is measurable in $Pr_{\Omega}(F)$ and continuous in $Pr_{X}(F)$. Ensure that the extension preserves the properties of measurability and continuity for the entire space. Probably some assumptions about the measurable space and the set $F$ are necessary. $(\Omega, \mathcal{F})$ can be a (perhaps complete) probability space.
Functions of this type are sometimes called Carathéodory functions or Random Functions.
Edit: I couldn't find anything about extending a function with two variables. But it is possible to find some results of extension for measurable functions of one variable. For example, Theorem 4.2.5 from Dudley's book or Proposition 3.3.4 from Srivastava's book.
So, an extra question would be, given that there are these results (the one that attracts me most is from Srivastava's book) of extending measurable functions, is it possible to apply to $f:F \to \mathbb{R}$? In other words, first extend it to a function In other words, first extend it to a measurable function $\hat{f}:\Omega \times Pr_X(F) \to \mathbb{R}$ and then continually extend to $\tilde{f}:\Omega \times X \to \mathbb{R}$.
