Total variation norm

The norm of total variation is a term from measure theory , a branch of mathematics that deals with the generalization of length and volume concepts. The total variation norm assigns a number to each signed measure and thus defines a norm on the vector space of the signed measures. The metric induced by the norm is then also called the total variation distance or total variation metric .

Sometimes there are also the terms total variation or total variation . However, these are ambiguous, they are also used for the measure composed of positive variation and negative variation , the variation of measure .

definition

Let there be a signed dimension on the measuring room and be the Jordan decomposition of the signed dimension and the Hahn decomposition of the base space as well as the variation of the signed dimension. Then is called ${\displaystyle \nu }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle \nu =\nu ^{+}-\nu ^{-}}$ ${\displaystyle \Omega =P\cup N}$ ${\displaystyle |\nu |}$

${\displaystyle \Vert \nu \Vert _{TV}:=\nu (P)-\nu (N)=\nu ^{+}(\Omega )+\nu ^{-}(\Omega )=|\nu |(\Omega )}$

the total variation norm of the signed measure . ${\displaystyle \nu }$

Examples

If the basic set is for a fixed one , then every finite signed measure can be defined by a vector ${\displaystyle \Omega =\{1,\dots ,n\}}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle x\in \mathbb {R} ^{n}}$

${\displaystyle \nu (A)=\sum _{i\in A}x_{i}}$.

The Hahn division would then be

${\displaystyle P=\{i\in \Omega \,|\,x_{i}>0\}{\text{ und }}N=\{i\in \Omega \,|\,x_{i}<0\}}$.

Accordingly, is the total variation of the signed measurement

${\displaystyle \Vert \nu \Vert _{TV}=\sum _{i=1}^{n}|x_{i}|=\Vert x\Vert _{1}}$

exactly the 1-norm of the vector.

If the signed dimension has a density with respect to a dimension , then is ${\displaystyle \nu }$ ${\displaystyle f}$ ${\displaystyle \mu }$

${\displaystyle \nu (A)=\int _{A}f\mathrm {d} \mu }$,

so the positive variation is given by and the negative variation by . So is ${\displaystyle \nu ^{+}(A)=\int _{A}\max(f,0)\mathrm {d} \mu }$ ${\displaystyle \nu ^{-}(A)=-\int _{A}\min(f,0)\mathrm {d} \mu }$

${\displaystyle \Vert \nu \Vert _{TV}=\int _{\Omega }\max(f,0)\mathrm {d} \mu -\int _{\Omega }\min(f,0)\mathrm {d} \mu =\int _{\Omega }|f|\mathrm {d} \mu }$.

properties

• The total variation norm turns the finite signed measures into a complete vector space that can even be arranged.
• As for every norm, a metric can be defined from the total variation norm

${\displaystyle d(\mu ,\nu )=\Vert \mu -\nu \Vert _{TV}}$.

This is called the total variation metric or total variation distance. These designations are used in particular when examining subsets of the signed measures that are not subspaces of the signed measures. Examples of this are the finite measures , the sub-probability measures and the probability measures .

• For finite measures on Borel's σ-algebra of a metric space , the weak convergence follows from the convergence with respect to the total variation distance .
• The total variation distance is equivalent to the Hellinger distance for probability measures with densities .

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 .