Editing Total variation (section)

====Total variation norm of complex measures====
If the measure <math>\mu</math> is [[Complex number|complex-valued]] i.e. is a [[complex measure]], its upper and lower variation cannot be defined and the Hahn&ndash;Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow {{Harvtxt|Rudin|1966|pp=137&ndash;139}} and define the total variation of the complex-valued measure <math>\mu</math> as follows

{{EquationRef|4|Definition 1.4.}} The '''variation''' of the complex-valued measure <math>\mu</math> is the [[set function]]

:<math>|\mu|(E)=\sup_\pi \sum_{A\isin\pi} |\mu(A)|\qquad\forall E\in\Sigma</math>

where the [[supremum]] is taken over all partitions <math>\pi</math> of a [[measurable set]] <math>E</math> into a countable number of disjoint measurable subsets.

This definition coincides with the above definition <math>|\mu|=\mu^++\mu^-</math> for the case of real-valued signed measures.