Lusin's Theorem

Introduction

Lusin's Theorem formalizes the measure-theoretic principle that pointwise convergence is "nearly" uniformly convergent. This is the second of Littlewood's famed three principles of measure theory, which he elaborated in his 1944 work "Lectures on the Theory of Functions"[1] as

"There are three principles, roughly expressible in the following terms: Every (measurable) set is nearly a finite sum of intervals; every function (of class Lp) is nearly continuous; every convergent sequence of functions is nearly uniformly convergent."

Lusin's theorem is hence a key tool in working with measurable functions, often enabling one to reduce measurable, yet intractible functions to the consideration of a continuous approximation.

Statement

Let $(\mathbb {R} ,{\mathcal {M}}_{\lambda },\lambda )$ be the Lebesque measure space on $\mathbb {R}$, and $E$ a measurable subset of $\mathbb {R}$ satisfying $\lambda (E)<+\infty .$ Let $f:E\to \mathbb {R}$ a be a measurable real-valued function on $E$. For all $\varepsilon >0,$ there exists a compact set $K\subseteq E$ such that the restriction of $f$ to $K$ is continuous.

Proof

Let $\varepsilon >0$ be given. We first show that if $A$ the existence of a compact set $K\subseteq E$ such that $\lambda (E\setminus K)<\varepsilon .$ From a standard theorem, we know already the existence of a closed set $V\subseteq E$ with $\lambda (E\setminus V)<\varepsilon /2.$ The sequence of measurable sets defined by $E_{n}=E\cap [-n,n]$ is nested and increasing, and satisfies $\bigcup _{n=1}^{\infty }E_{n}=E.$ By continuity of the measure from below, we deduce the existence of an $N\in \mathbb {N}$ such that $\lambda (E\setminus [-n,n])<\varepsilon /2./math>Then<math>K:=E\cap [-N,N]\subseteq E$ is a closed subset of the compact set $[-N,N],$ and is hence compact. Further, measurability implies the desired inequality $\lambda (E\setminus K)<\varepsilon .$

By cardinality considerations, we may enumerate the collection of open intervals in $\mathbb {R}$ with rational endpoints, say by Failed to parse (syntax error): {\displaystyle \{V_n\}_{n\in \mathbb{N}. }

References

[1] Littlewood, J. E. "Lectures on the Theory of Functions." Oxford University Press. 1944. [2] Talvila, Erik; Loeb, Peter. "Lusin's Theorem and Bochner Integration." arXiv. 2004. https://arxiv.org/abs/math/0406370 [3] https://conf.math.illinois.edu/~mjunge/54004-lusin.pdf