I'm a beginner in learning Measure theory. I would like to ask are there some common problem solving tricks in Measure theory.
For example, usually when I'm doing a measure theory problem, once I rewrite a set into simpler sets, the question is basically solved. So I guess an example trick would be "measure theory problems are mostly about rewriting sets".
Also, the following seems to be a general procedure for proving integral identities/theorems:
- prove it for indicators $1_A$
- linearity proves it for simple functions
- monotone convergence theorem proves it for non-negative measurable functions
Are there any other tricks? Also, are my tricks common/useful? Many thanks in advance.