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Published in:
2018 | OriginalPaper | Chapter
3. Measurable Functions
Author : Vladimir Kadets
Published in: A Course in Functional Analysis and Measure Theory
Publisher: Springer International Publishing
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Abstract
Let \(({\Omega _1 ,\Sigma _1})\) and \(({\Omega _2 ,\Sigma _2})\) be sets endowed with \(\sigma \)-algebras of subsets. A function \(f:\Omega _1 \rightarrow \Omega _2\) is said to be measurable if \(f^{- 1} (A) \in \Sigma _1\) for all \(A \in \Sigma _2\). Measurable functions play in measure theory the same role that continuous functions do in the theory of topological spaces. In this chapter we present the basic properties of measurable functions, the Lebesgue’s approximation of measurable functions by simple ones, and the relationship between various types of convergence of sequences of measurable functions.