Total Variation

Why

We want a norm on the space of measures.

Definition

The total variation of finite measure is the variation measure of the base set. We show below that the total variation is a norm on the vector space of finite measures.

Notation

Let $(X, \mathcal{A} )$ be a measurable space and $\mu : X \to \R $ be a finite signed measure. We denote the total variation by $\tvnorm{\mu }$.

Let $\vmeas{\mu }$ be the variation of $\mu $. Then, $\tvnorm{\mu } = \vmeas{\mu }(X)$.