Let $ (X,\mu) $ be a standard measure space - so that we may assume that $X$ is the unit interval $[0,1]$ with the Borel $\sigma$-algebra. Consider $X \times X$ with the product measure $\mu \times \mu $ defined on the product $\sigma$-algebra.
Let $f$ and $g$ be two functions defined on $X \times X$ such that:
- $f$ is measurable.
- For a.e. $x$, the function $y \to g(x,y)$ is measurable.
- For every measurable subset $E\subset X \times X$, we have $$\int \chi _E \cdot f(x,y) d\mu(y) = \int \chi_E \cdot g(x,y) d\mu(y) $$ for a.e. $x$.
Is this data sufficient to imply that $g$ is measurable as well? and that $f=g$ a.e.?
This is a (hopefully more interesting) variant of my previous question Measurable functions on product measures, which was answered in the negative by @Nate Eldredge (but the same example does not work in this case).