Lebesgue's decomposition theorem

In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures and on a measurable space there exist two σ-finite signed measures and such that: (that is, is absolutely continuous with respect to ) (that is, and are singular). These two measures are uniquely determined by and Refinement Lebesgue's decomposition theorem can be refined in a number of ways.