When we type an answer, we need sometimes classical facts, which will be long to expand and are well-known.
For example, a bounded measurable function can be approached uniformly by simple functions, or the fact that convergence in $L^p$ implies convergence almost everywhere of a sub-sequence.
I suggest to write in the answers a collection of these facts by fields (like "measure-theory", "real-analysis", etc...) including the fact and a link on the site to a question which deals with it. Since these fields are far from being disjoint, a same fact could belong to several answers.
The point is that if someone is looking for a proof of a fact, he/she will include a link in his/her answer after having found it in this page.