If μ is a real measure (i.e., a measure that takes on real values), then one can decompose it according to where it is positive and negative. The positive variation is defined by μ^+ = 1/2( left bracketing bar μ right bracketing bar + μ), where left bracketing bar μ right bracketing bar is the total variation. Similarly, the negative variation is μ^- = 1/2( left bracketing bar μ right bracketing bar - μ). Then the Jordan decomposition of μ is defined as μ = μ^+ - μ^-. When μ already is a positive measure then μ = μ^+.

measure | polar representation | total variation