The Lebesgue Integral

Edwin Hewitt and Karl Stromberg
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Edwin Hewitt: The University of Washington
Karl Stromberg: The University of Oregon

Chapter Chapter Three in Real and Abstract Analysis, 1965, pp 104-187 from Springer

Abstract: Abstract Integration from one point of view is an averaging process for functions, and it is in this spirit that we will introduce and discuss integration. In applying an averaging process to a class $${\mathcal{F}}$$ real- or complex-valued functions, a number I(f) is assigned to each $$f \in {\mathcal{F}}$$ . If I(f) is to be an average, then it should certainly satisfy the conditions $$\matrix{{I(f + g) = I(f) + I(g),} \cr {I(\alpha f) = \alpha I(f)} \cr }$$ for f, $$g \in {\mathcal{F}}$$ and α ∈R. A less essential but often desirable property for I is that I(f) ≧ 0 if f ≧ 0. In some cases these three properties suffice to identify the averaging process completely.

Keywords: Measure Space; Compact Hausdorff Space; Outer Measure; Nondecreasing Sequence; Finite Measure Space (search for similar items in EconPapers)
Date: 1965
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DOI: 10.1007/978-3-642-88044-5_3

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