The Functional Monotone Class Theorem

The monotone class theorem is a very helpful and frequently used tool in measure theory. As measurable functions are a rather general construct, and can be difficult to describe explicitly, it is common to prove results by initially considering just a very simple class of functions. For example, we would start by looking at continuous or piecewise constant functions. Then, the monotone class theorem is used to extend to arbitrary measurable functions. There are different, but related, `monotone class theorems’ which apply, respectively, to sets and to functions. As the theorem for sets was covered in a previous post, this entry will be concerned with the functional version. In fact, even for the functional version, there are various similar, but slightly different, statements of the monotone class theorem. In practice, it is beneficial to use the version which most directly applies to the specific application. So, I will state and prove several different versions in this post. Continue reading “The Functional Monotone Class Theorem”

The Monotone Class Theorem

The monotone class theorem, and closely related {\pi}-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 {{\mathbb R}}, 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”