Originally published at 狗和留美者不得入内. You can comment here or there.

Let be measurable functions on measure space . converges to globally in measure if for every ,

.

To see that this means the existence of a subsequence with pointwise convergence almost everywhere, let be such that for , , with increasing. (We invoke the definition of limit here.) If we do not have pointwise convergence almost everywhere, there must be some such that there are infinitely many such that . There is no such for the subsequence as .

This naturally extends to every subsequence’s having a subsequence with pointwise convergence almost everywhere (limit of subsequence is same as limit of sequence, provided limit exists). To prove the converse, suppose by contradiction, that the set of , for which there are infinitely many such that for some has positive measure. Then, there must be infinitely many such that is satisfied by a positive measure set. (If not, we would have a countable set in for bad points, whereas there are uncountably many with infinitely bad points.) From this, we have a subsequence without a pointwise convergent subsequence.