total variation measure

Given a complex measure μ, there exists a positive measure denoted left bracketing bar μ right bracketing bar which measures the total variation of μ, also sometimes called simply "total variation." In particular, left bracketing bar μ right bracketing bar (E) on a subset E is the largest sum of "variations" for any subdivision of E. Roughly speaking, a total variation measure is an infinitesimal version of the absolute value. More precisely, left bracketing bar μ right bracketing bar (E) = sup sum_i left bracketing bar μ(E_i) right bracketing bar where the supremum is taken over all partitions union E_i of E into measurable subsets E_i.

Jordan measure decomposition | measure | polar representation | Riesz representation theorem