Editing Total variation (section)

===Total variation of a measure===
The total variation is a [[norm (mathematics)|norm]] defined on the space of measures of bounded variation.  The space of measures on a σ-algebra of sets is a [[Banach space]], called the [[ca space]], relative to this norm.  It is contained in the larger Banach space, called the [[ba space]], consisting of ''[[Finitely additive measure|finitely additive]]'' (as opposed to countably additive) measures, also with the same norm. The [[distance function]] associated to the norm gives rise to the total variation distance between two measures ''μ'' and ''ν''.

For finite measures on '''R''', the link between the total variation of a measure ''μ'' and the total variation of a function, as described above, goes as follows. Given ''μ'', define a function <math>\varphi\colon \mathbb{R}\to \mathbb{R}</math> by
:<math>\varphi(t) = \mu((-\infty,t])~.</math>
Then, the total variation of the signed measure ''μ'' is equal to the total variation, in the above sense, of the function <math>\varphi</math>. In general, the total variation of a signed measure can be defined using [[Hahn decomposition theorem|Jordan's decomposition theorem]] by
:<math>\|\mu\|_{TV} = \mu_+(X) + \mu_-(X)~,</math>
for any signed measure ''μ'' on a measurable space <math>(X,\Sigma)</math>.