Let $(X, \mathcal X)$ and $(Y, \mathcal Y)$ be measurable spaces, and let $f$ be a measurable function from $X$ into $Y$. Assume that the graph of $f$, i.e. $Gr(f) = \{(x,y): y = f(x)\}$, is $\mathcal X \otimes \mathcal Y$ measurable.
I am wondering if there anything interesting can be said about the measurability of the graph if $\mathcal Y$ is the largest sigma-algebra for which $f$ is measurable. For instance,
Conjecture. $\mathcal Y$ is the largest sigma-algebra for which $f$ is measurable if and only if for every sigma-algebra $\mathcal A$ for which $Gr(f)$ is $\mathcal X \otimes \mathcal A$ measurable, $\mathcal Y \subseteq \mathcal A$.
Is this true? Or is one direction true?
Now I believe the conjecture is false, though the argument is complete.
By the result mentioned here, $Gr(f) \in \mathcal X \otimes \mathcal A$ iff $\mathcal A$ contains a countably generated sub-sigma-algebra that contains the singletons. If $\mathcal Y$ is not countably generated and $Gr(f)\in\mathcal X \otimes \mathcal Y$, then we can find a proper sub-sigma-algebra $\mathcal A$ of $\mathcal Y$ such that $Gr(f) \in \mathcal X \otimes \mathcal A$. It remains to show that such a $\mathcal Y$ can be found that is also the largest algebra with respect to which $f$ is measurable.
