 TOPICS # Total Variation

Given a complex measure , there exists a positive measure denoted which measures the total variation of , also sometimes called simply "total variation." In particular, on a subset is the largest sum of "variations" for any subdivision of . Roughly speaking, a total variation measure is an infinitesimal version of the absolute value.

More precisely, (1)

where the supremum is taken over all partitions of into measurable subsets .

Note that may not be the same as . When already is a positive measure, then . More generally, if is absolutely continuous, that is (2)

then so is , and the total variation measure can be written as (3)

The total variation measure can be used to rewrite the original measure, in analogy to the norm of a complex number. The measure has a polar representation (4)

with .

Jordan Measure Decomposition, Measure, Polar Representation, Riesz Representation Theorem

This entry contributed by Todd Rowland

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Rowland, Todd. "Total Variation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TotalVariation.html