Abstract
A measure in the extended sense, or just a measure*, is a non-negative, countably additive, ℝ* valued function μ on a δ-ring A with μ(∅) = 0. The function on A that is 0 at ∅ and ∞ elsewhere is a measure*, each measure is a measure*, and each finite valued measure* is a measure. Classical Lebesgue measure for ℝ (see note 4.13 (i)) is the prototypical example of a measure*. A function f is integrable (or integrable*) w.r.t. a measure* μ on A iff it is integrable (integrable*) w.r.t. the measure μ 0 = μ|{A: A ∈Aand μ(A) < ∞} and in this case ∫ f dμ = ∫ f dμ 0 . Thus the integral w.r.t. classical Lebesgue measure is indentical with the integral w.r.t. Λ1.
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© 1988 Springer-Verlag New York Inc.
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Kelley, J.L., Srinivasan, T.P. (1988). Measures* and Mappings. In: Measure and Integral. Graduate Texts in Mathematics, vol 116. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4570-4_9
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DOI: https://doi.org/10.1007/978-1-4612-4570-4_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8928-9
Online ISBN: 978-1-4612-4570-4
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