The Radon-Nikodym Theorem

Signed measures andHahn decompositionA signed measure ν on a measurable space(X,M) is an extended real-valued set function νdefined for the sets of M and satisfying thefollowing conditions:ν assumes at most one of the values +∞, -∞.ν(∅) = 0.ν(⋃ i≥1 E i ) = ∑ i≥1 ν(E i ) for any sequence (E i ) ofdisjoint measurable sets.