Editing Total variation (section)

====Modern definition of total variation norm====
{{Harvtxt|Saks|1937|p=11}} uses upper and lower variations to prove the [[Hahn decomposition theorem|Hahn&ndash;Jordan decomposition]]: according to his version of this theorem, the upper and lower variation are respectively a [[non-negative]] and a [[non-positive]] [[Measure (mathematics)|measure]]. Using a more modern notation, define

:<math>\mu^+(\cdot)=\overline{\mathrm{W}}(\mu,\cdot)\,,</math>
:<math>\mu^-(\cdot)=-\underline{\mathrm{W}}(\mu,\cdot)\,,</math>

Then <math>\mu^+</math> and <math>\mu^-</math> are two non-negative [[measure (mathematics)|measure]]s such that

:<math>\mu=\mu^+-\mu^-</math>
:<math>|\mu|=\mu^++\mu^-</math>

The last measure is sometimes called, by [[abuse of notation]], '''total variation measure'''.