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 $g:\mathbb{R}\to \mathbb{R}$, $\forall ϵ>0$ there is a closed set $F\subseteq \mathbb{R}$ such that $|X\backslash F|<ϵ$ and $g{|}_F$ is a continuous function on $F$.

Note that this doesn’t mean that $g$ is continuous on $\mathbb{R}$ at all the points in $F$, it means that when considering only the domain as restricted to $F$ then $g$ is continuous.

Proof

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

Theorem (Version 2)

For $E\subseteq R$ and a Borel measurable function $g:E\to \mathbb{R}$, for every $ϵ>0$ there exists a closed set $F\subseteq E$ and a continuous function $h:\mathbb{R}\to \mathbb{R}$ such that $|E\backslash F|<ϵ$ and $g{|}_F=h{|}_F$

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