If $(f_n)_n\rightarrow f$ almost everywhere and in measure, does $(f_n)_n \rightarrow f$ almost uniformly?
I've been wondering about this question and came with no result. I know that convergence almost everywhere does not imply convergence almost everywhere, but I am not aware of any counterexample if the extra condition of convergence in measure is added.
The other implication is true. If $(f_n)_n \rightarrow f$ almost uniformly, then it is correct that it converges for $f$ both almost everywhere and in measure, which makes me think that the converse (the question) is not true. Thank you!