Two complex measures μ and ν on a measure space X, are mutually singular if they are supported on different subsets. More precisely, X = A union B where A and B are two disjoint sets such that the following hold for any measurable set E, 1. The sets A intersection E and B intersection E are measurable. 2. The total variation of μ is supported on A and that of ν on B, i.e., left double bracketing bar μ right double bracketing bar (B intersection E) = 0 = left double bracketing bar ν right double bracketing bar (A intersection E). The relation of two measures being singular, written as μ⊥ν, is plainly symmetric. Nevertheless, it is sometimes said that "ν is singular with respect to μ."

absolutely continuous | complex measure | Lebesgue decomposition | Lebesgue measure