Summary
20.1 Let μ be a Radon measure on a locally compact space X. Let Y be a locally compact subspace of X. Integration with respect to the induced measure μ Y is done in the most natural way (Theorem 20.1.1). Piecing together Radon measures, as done in Theorem 20.1.2, is extremely useful in analysis and differential geometry.
20.2 Let μ be a Radon measure on a locally compact space T. A function g from T into C is said to be locally integrable when gh is μ-integrable for every continuous function h with compact support. In this case gμ: h↦∫ghdμ is a Radon measure. We study how to integrate with respect to gμ (Theorems 20.2.1 and 20.2.2).
20.3 The Radon-Nikodym theorem and Lebesgue’s decomposition theorem still hold for Radon measures.
20.4 From Chapter 10 follow some results on the duality of L p C (μ) spaces, when μ is a Radon measure.
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© 1996 Springer-Verlag New York, Inc.
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Simonnet, M. (1996). Radon Measures Defined by Densities. In: Measures and Probabilities. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4012-9_20
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DOI: https://doi.org/10.1007/978-1-4612-4012-9_20
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