Convergence almost sure pointless?

Question

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a uniform convergence almost sure due to Egorov's theorem. So why do people in probability theory refer to this weaker concept?

If anything is unclear, please let me know.

Answer 1

It is useful to operate with a pair of concepts, such as a.e. convergence and almost uniform convergence, of which one is visibly weaker but in fact implies the other. When you use this property, you can use the stronger one. When you have to prove that it holds, you will want to prove the weaker one.

Given a sequence $X_n$, would you rather prove that they converge a.e., or that they converge almost uniformly. For a.e., the proof naturally begins with

Suppose $X_n$ fail to converge a.e. Then there exists a set $E$ of positive measure on which convergence fails everywhere. From this set we select a subset such that ... ... and arrive at a contradiction.

Having this concrete set $E$ on which things go badly focuses the effort in the above proof.

For almost uniform convergence, what would you do?

Suppose $X_n$ fail to converge almost uniformly. Then there is $\epsilon>0$ such that for every set $E$ with $P(E)<\epsilon$ the sequence fails to converge uniformly on the complement of $E$. Hmm....

You don't get much to work with here.