The monotone class theorem, and closely related -system lemma, are simple but fundamental theorems in measure theory, and form an essential step in the proofs of many results. General measurable sets are difficult to describe explicitly so, when proving results in measure theory, it is often necessary to start by considering much simpler sets. The monotone class theorem is then used to extend to arbitrary measurable sets. For example, when proving a result about Borel subsets of , we may start by considering compact intervals and then apply the monotone class theorem. I include this post on the monotone class theorem for reference. Continue reading “The Monotone Class Theorem”