I have confronted the following question while solving another problem. Specifically, in my case, I'm curious whether some set of measure zero in $\mathbb{R}^{k+n}$ be measure zero in $\mathbb{R}^n$.
In order to solve the below problem, I think I need this result.
The way I thought to solve this problem is, first given $D$, a set of measure zero in $R^k$, and a fixed $x_0\in A-D$, $D\times B$ is also of measure zero in $R^{k+n}$, and since $f$ is integrable in $Q$, the set of discontinuities, say $E$ is also of measure zero. Then clearly, $f(x_0,y)$, for all $y\in B$, is continuous except for some set of measure zero in $R^{k+n}$. Hence, the statement in the question is proven if I can show that, $f(x_0,y)$, for all $y\in B$, is continuous almost everywhere in $R^n$. But would this be true? And if so how can I prove it?
I would greatly appreciate any suggestions or solutions.