I know that convergence almost everywhere implies convergences in measure(sometime it said locally). I also know that convergence in measure implies the existence of subsequence which is convergent almost everywhere.
However I wonder if there are some additional conditions from which the convergence in measure will imply convergence almost everywhere (for a whole sequence, not just subsequence).
Please don't mention discreteness. Let's just take real line for example, and a sequence of functions on it.
Thank you, for any help.