Lusin's Theorem

Introduction

Lusin's Theorem formalizes the measure-theoretic principle that pointwise convergence is "nearly" uniformly convergent. This is the third 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 sequences of measurable functions, often allowing one to globally deduce the strong conclusions resulting from uniform convergence to the relatively weak assumption of a pointwise convergent sequence.

Classical Statement

Let $(\mathbb {R} ,{\mathcal {M}}_{\lambda },\lambda )<math>betheLebesquemeasurespaceon<math>\mathbb {R}$, and $E$ a measurable subset of $\mathbb {R}$.

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