Luzin’s Theorem

Overview

Any Borel measurable function is “almost” continuous. If the function is not continuous, we can restrict the function to a closed set where the measure of everything not included is arbitrarily small.

There are two versions both say basically the same thing

Theorem (Version 1)

For a Borel measurable function , there is a closed set such that and is a continuous function on .

Note that this doesn’t mean that is continuous on at all the points in , it means that when considering only the domain as restricted to then is continuous.

Proof

I didn’t want to copy it again, so its on @axler2019 page 66

Theorem (Version 2)

For and a Borel measurable function , for every there exists a closed set and a continuous function such that and

I personally prefer this version. Note that h is called a continuous extension of g.

Proof

I also don’t want to write this one out. It’s on page 68 of @axler2019