approximation by simple functions

Overview

Any measurable function can be approximated “from below” by a measureable simple function. This is nice since simple functions can be much easier to work with.

Theorem (if you can call it that)

For any measurable function on a measurable space , then there exists a sequence of measurable simple functions such that

1. for all and all Or in other words, f is an ‘increasing’ sequence of functions.

2. for all

3. If f is bounded, then converges uniformly on X

Proof

The general idea is that you can split the domain up into (a lot) of finitely intervals which get smaller and closer to having the actual function value.

First set bounds for the function. For a set . Let if and if

For anything else inside those bounds (of the domain) we split each interval into subintervals. is defined as the left endpoint of the subinterval of the domian that it is found in.

Formally this can be written as