Abstract
If a function f is summable over a measure μ, then, according to Theorem 3.5.4, the charge ρ(E) = ∫ E fdμ is absolutely continuous with respect to the measure μ. Here, we give a more convenient definition of absolute continuity and prove the Radon-Nikodym theorem that allows one to represent every absolutely continuous charge in the indicated form. This theorem enables one to introduce the notion of the Radon-Nikodym derivative and to prove the theorem on the change of variables in the Lebesgue integral by using this notion. The Radon-Nikodym theorem also allows one to give a simple proof of the Lebesgue theorem on the representation of an arbitrary charge in the form of a sum of absolutely continuous and singular charges. These results are then used in the investigation of functions of bounded variation.
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© 1996 Birkhäuser Verlag
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Berezansky, Y.M., Sheftel, Z.G., Us, G.F. (1996). Absolute Continuity and Singularity of Measures, Charges, and Functions. Radon-Nikodym Theorem. Change of Variables in the Lebesgue Integral. In: Functional Analysis. Operator Theory Advances and Applications, vol 85. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9185-1_5
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DOI: https://doi.org/10.1007/978-3-0348-9185-1_5
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9939-0
Online ISBN: 978-3-0348-9185-1
eBook Packages: Springer Book Archive