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.

Classical 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 Failed to parse (unknown function "\varepsilone"): {\displaystyle \varepsilone>0, } there exists a compact set $K\subseteq \mathbb {R}$

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