Absolute Continuity and Singularity of Measures, Charges, and Functions. Radon-Nikodym Theorem. Change of Variables in the Lebesgue Integral

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.