Editing Total variation (section)

====Total variation norm of vector-valued measures====

The variation so defined is a [[positive measure]] (see {{Harvtxt|Rudin|1966|p=139}}) and coincides with the one defined by {{EquationNote|3|1.3}} when <math>\mu</math> is a [[signed measure]]: its total variation is defined as above. This definition works also if <math>\mu</math> is a [[vector measure]]: the variation is then defined by the following formula

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

where the supremum is as above. This definition is slightly more general than the one given by {{Harvtxt|Rudin|1966|p=138}} since it requires only to consider ''finite partitions'' of the space <math>X</math>: this implies that it can be used also to define the total variation on [[Sigma additivity|finite-additive measures]].