Editing Total variation (section)

==Definitions==

===Total variation for functions of one real variable===
{{EquationRef|1|Definition 1.1.}} The '''total variation''' of a [[real number|real]]-valued (or more generally [[complex number|complex]]-valued) [[function (mathematics)|function]] <math>f</math>, defined on an [[interval (mathematics)|interval]] <math> [a , b] \subset \mathbb{R}</math> is the quantity

:<math> V_a^b(f)=\sup_{\mathcal{P}} \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |, </math>

where the [[supremum]] runs over the [[Set (mathematics)|set]] of all [[partition of an interval|partitions]] <math> \mathcal{P} = \left\{P=\{ x_0, \dots , x_{n_P}\} \mid P\text{ is a partition of } [a,b] \right\} </math> of the given [[interval (mathematics)|interval]]. Which means that <math>a = x_{0} < x_{1} <  ...  < x_{n_{P}} = b</math>.

===Total variation for functions of ''n'' > 1 real variables ===
{{citation needed section|date=September 2022}}
{{EquationRef|2|Definition 1.2.}}<ref name="10.1093/oso/9780198502456.001.0001">{{cite book |last1=Ambrosio |first1=Luigi |last2=Fusco |first2=Nicola |last3=Pallara |first3=Diego |title=Functions of Bounded Variation and Free Discontinuity Problems |date=2000 |publisher=Oxford University Press |isbn=9780198502456 |url=https://doi.org/10.1093/oso/9780198502456.001.0001}|pages=119|doi=10.1093/oso/9780198502456.001.0001 }}</ref> Let '''Ω''' be an [[open subset]] of '''R'''<sup>''n''</sup>. Given a function ''f'' belonging to ''L''<sup>1</sup>('''Ω'''), the '''total variation''' of ''f'' in '''Ω''' is defined as

:<math> V(f,\Omega):=\sup\left\{\int_\Omega f(x) \operatorname{div} \phi(x) \, \mathrm{d}x \colon \phi\in  C_c^1(\Omega,\mathbb{R}^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\right\}, </math>

where  
* <math> C_c^1(\Omega,\mathbb{R}^n)</math> is the [[Set (mathematics)|set]] of [[Smooth function|continuously differentiable]] [[vector-valued function|vector functions]] of [[support (mathematics)#Compact support|compact support]] contained  in <math>\Omega</math>, 
* <math> \Vert\;\Vert_{L^\infty(\Omega)}</math> is the [[essential supremum]] [[Norm (mathematics)|norm]], and 
* <math>\operatorname{div}</math> is the [[divergence]] operator. 
This definition ''does not require'' that the [[Domain of a function|domain]] <math>\Omega \subseteq \mathbb{R}^n</math> of the given function be a [[bounded set]].

===Total variation in measure theory===

====Classical total variation definition====
Following {{Harvtxt|Saks|1937|p=10}}, consider a [[signed measure]] <math>\mu</math> on a [[sigma-algebra|measurable space]] <math>(X,\Sigma)</math>: then it is possible to define two [[set function]]s <math>\overline{\mathrm{W}}(\mu,\cdot)</math> and <math>\underline{\mathrm{W}}(\mu,\cdot)</math>, respectively called '''upper variation''' and '''lower variation''', as follows

:<math>\overline{\mathrm{W}}(\mu,E)=\sup\left\{\mu(A)\mid A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>
:<math>\underline{\mathrm{W}}(\mu,E)=\inf\left\{\mu(A)\mid A\in\Sigma\text{ and }A\subset E \right\}\qquad\forall E\in\Sigma</math>

clearly

:<math>\overline{\mathrm{W}}(\mu,E)\geq 0 \geq \underline{\mathrm{W}}(\mu,E)\qquad\forall E\in\Sigma</math>

{{EquationRef|3|Definition 1.3.}} The '''variation''' (also called '''absolute variation''') of the signed measure <math>\mu</math> is the set function

:<math>|\mu|(E)=\overline{\mathrm{W}}(\mu,E)+\left|\underline{\mathrm{W}}(\mu,E)\right|\qquad\forall E\in\Sigma</math>

and its '''total variation''' is defined as the value of this measure on the whole space of definition, i.e.

:<math>\|\mu\|=|\mu|(X)</math>

====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'''.

====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.

====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]].

====Total variation of probability measures====
{{unreferenced section|date=May 2012}}
{{main|Total variation distance of probability measures}}
The total variation of any [[probability measure]] is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are [[probability measure]]s, the '''[[total variation distance of probability measures]]''' can be defined as <math>\| \mu - \nu \|</math> where the norm is the total variation norm of signed measures.  Using the property that <math>(\mu-\nu)(X)=0</math>, we eventually arrive at the equivalent definition

:<math>\|\mu-\nu\| = |\mu-\nu|(X)=2 \sup\left\{\,\left|\mu(A)-\nu(A)\right| : A\in \Sigma\,\right\}</math>

and its values are non-trivial. The factor <math>2</math> above is usually dropped (as is the convention in the article [[total variation distance of probability measures]]). Informally, this is the largest possible difference between the probabilities that the two [[probability distribution]]s can assign to the same event. For a [[categorical distribution]] it is possible to write the total variation distance as follows

:<math>\delta(\mu,\nu) = \sum_x \left| \mu(x) - \nu(x) \right|\;.</math>

It may also be normalized to values in  <math>[0, 1]</math> by halving the previous definition as follows

:<math>\delta(\mu,\nu) = \frac{1}{2}\sum_x \left| \mu(x) - \nu(x) \right|</math><ref>{{cite web|last1=Gibbs|first1=Alison|author2=Francis Edward Su|title=On Choosing and Bounding Probability Metrics|url=https://www.math.hmc.edu/~su/papers.dir/metrics.pdf|access-date=8 April 2017|pages=7|date=2002}}</ref>