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Published in:
2017 | OriginalPaper | Chapter
5. Measurable Functions
Author : William P. Ziemer
Published in: Modern Real Analysis
Publisher: Springer International Publishing
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Abstract
The class of measurable functions will play a critical role in the theory of integration. It is shown that this class remains closed under the usual elementary operations, although special care must be taken in the case of composition of functions. The main results of this chapter are the theorems of Egorov and Lusin. Roughly, they state that pointwise convergence of a sequence of measurable functions is “nearly” uniform convergence and that a measurable function is “nearly” continuous.