Mutually singular measures

Summary

Two measures on the same space are mutually singular when they live on disjoint parts of the space. One of them gives zero mass to a set, and the other gives zero mass to its complement. Mutual singularity is the opposite extreme of absolute continuity of measures. The Lebesgue decomposition theorem splits a σ-finite measure, relative to a σ-finite reference measure, into an absolutely continuous part and a singular part.