Abstract
A real (finite) valued function f on X is locally μ integrable iff μ is a measure on a δ-ring A of subsets of X and f χ A ∈ L 1 (μ)for all A in A. In this case f.μ, the indefinite integral of f with respect to μ, is the function A↦∫ A A for A in A. This function is always countably additive and hence f.μ is a measure if f is non-negative. Consequently f.μ is the difference of two measures, f +.μ and f-.μ. These two measures have little to do with each other: one of them “lives on” the set {x:f(x) ≧ 0} and the other lives on {x:f(x) < 0}.
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© 1988 Springer-Verlag New York Inc.
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Kelley, J.L., Srinivasan, T.P. (1988). Signed Measures and Indefinite Integrals. In: Measure and Integral. Graduate Texts in Mathematics, vol 116. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4570-4_10
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DOI: https://doi.org/10.1007/978-1-4612-4570-4_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8928-9
Online ISBN: 978-1-4612-4570-4
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