Joint Measurability under Indistinguishability

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space and $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces.

Consider a $\mathcal{F}\otimes \mathcal{A}–\mathcal{B}$-measurable function $f\colon \Omega\times X\to Y$ (call such functions jointly measurable).

Let $g$ be another function $\Omega\times X\to Y$, but let $g$ only be measurable in the following sense to begin with: For all $x\in X$ the position $(\omega\mapsto g(\omega, x)$ is $\mathcal{F}-\mathcal{B}$-measurable.

Now, does it follow from the indistinguishability of $f$ and $g$, i.e. $$f(\omega, x) = g(\omega, x),\quad x\in X$$ for almost all $\omega\in\Omega$, that $g$ is jointly measurable as well?

Answer 1

Let $\mathcal{N}$ be the set of $\omega\in\Omega$ upon which $f$ differs from $g$ ($\mathcal{N}$ is thus negligible wrt $\mathbb{P}$). That is $f = g$ except on the set $\mathcal{N}\times X$. Thus you have for $C\in\mathcal{B}$ $$ g^{-1}(C) = \left(g^{-1}(C)\cap(\mathcal{N}\times X)\right) \cup \left(g^{-1}(C)\cap(\mathcal{N}\times X)^{c}\right) $$ Given that $f = g$ on the set $(\mathcal{N}\times X)^{c}$ one has: $$ g^{-1}(C) = \left(g^{-1}(C)\cap(\mathcal{N}\times X)\right) \cup \left(f^{-1}(C)\cap(\mathcal{N}\times X)^{c}\right) $$ The term $\left(f^{-1}(C)\cap(\mathcal{N}\times X)^{c}\right)$ clearly belongs to $\mathcal{F}\otimes \mathcal{A}$. However, the term $g^{-1}(C)\cap(\mathcal{N}\times X)$ does not necessarily belong to $\mathcal{F}\otimes \mathcal{A}$. In the case where the space $(X, \mathcal{A})$ can be endowed with a finite measure $\mu$, then the set $\mathcal{N}\times X$ is also negligible wrt $\mathbb{P}\otimes\mu$ and one can then complete this space with sets negligible wrt to this measure to obtain measurability of g.