Total Variation

Given a complex measure mu, there exists a positive measure denoted |mu| which measures the total variation of mu, also sometimes called simply "total variation." In particular, |mu|(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,


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

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


then so is |mu|, and the total variation measure can be written as


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


with |h|=1.

See also

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

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Total Variation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications