Suppose that we have a sequence of probability densities $f_{n}(x,y)$ on a pair of random variables $(X,Y)$ that converges pointwise almost everywhere to a probability density $f(x,y)$ with respect to a common dominating measure $\mu$. Is it necessarily the case that the two sequences of marginal densities $(f_{n}(x))_{n}$ and $(f_{n}(y))_{n}$ converge pointwise almost everywhere to $f(x)$ and $f(y)$ respectively?
Clearly, this can be made to hold under domination/uniform integrability conditions, but I'd be interested to know if there are counterexamples if no further assumptions are made.
Some additional thoughts: by Scheffé's lemma, we know that the induced probability measures $P_{n}$ converge in total variation to the distribution $P$ induced by $f$. We can then deduce that the marginal distributions also converge in total variation, so there certainly exist subsequences $(f_{n_{j}}(x))$ and $(f_{n_{k}}(y))$ that converge pointwise a.e. to $f(x)$ and $f(y)$ respectively. Thus, if the assertion is true, it would suffice to show that the sequences $(f_{n}(x))_{n}$ and $(f_{n}(y))_{n}$ converge pointwise a.e. Conversely, any counterexample must show that the sequences do not converge pointwise a.e.