Luzin's theorem

• categories: Measure Theory, Theorem

Luzin’s theorem relates measurable functions and continuous functions by stating that any measurable function can be approximated by continuous functions, except on a set of arbitrarily small measure.

Statement:

Let $f$ be a measurable function defined on a set $E\subset {\mathbb{R}}^n$ with finite measure, i.e., $\mu (E)<\infty$. For every $ϵ>0$, there exists a closed set $F_{ϵ}\subseteq E$ such that:

• $\mu (E\setminus F_{ϵ})<ϵ$.
• The restriction of $f$ to $F_{ϵ}$ is continuous.

Intuition:

Luzin’s theorem shows that any measurable function behaves “almost” like a continuous function. More precisely, for any small $ϵ$, the function can be made continuous on most of its domain, except for a small set of measure less than $ϵ$.