Integral of Nonnegative Functions

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro, Electrical Engineering Department

Chapter 3 in Essentials of Measure Theory, 2015, pp 41-55 from Springer

Abstract: Abstract Let X be an arbitrary set. A simple function on X is a real-valued function $$\varphi: X\! \rightarrow \mathbb{R}$$ with a finite range (i.e., a function that takes on only a finite number of distinct values). It is clear that $$\varphi$$ is a simple function if and only if it can be represented as a linear combination of characteristic functions, $$\displaystyle{\varphi \,=\sum _{ i=1}^{n}\alpha _{ i}^{_{\chi }}{}_{ E_{i}},}$$

Keywords: Measurable Function; Measure Space; Simple Function; Nonnegative Function; Canonical Representation (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-22506-7_3

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