Given a sequence of functions $f_{n,m}:\mathbb{R}^{n} \to \mathbb{R}$, suppose that $$\displaystyle\lim_{m} f_{n,m}(x)$$ exists almost everywhere (for any fixed n) and also suppose that $$\displaystyle\lim_{n} f_{n,m}$$ exists in norm $L^p$(for any fixed $m$).
I would like to know if the iterative limit commute, that is, if $$\displaystyle\lim_{n}\lim_{m} f_{n,m}(x) = \displaystyle\lim_{m}\lim_{n} f_{n,m}(x)$$ almost everywhere (where the limit in the index $n$ is again understood as a $L^p$ limit and the limit in the index $m$ is understood as a limit almost everywhere).
If not, which are the sufficient conditions for this to happen? Thanks in advance.