The Radon-Nikodym Theorem and its Extension to Signed Measures

Abstract

Given two measures and  on a measurable space (X,  ), a natural question that comes out is that if one can represent in terms of  via some linear operator. The Radon-Nikodym Theorem states that it is possible, under some hypothesis, to find a representation via the integral operator, that, given a measurable space (X, ), if  a  finite measure which is absolutely continuous with respect to a  finite measure  on(X), then there is a non-negative measurable function f on X such that   (E) = for any measurable set E.