Singular measure

A singular measure is a term from the mathematical branch of measure theory . It plays a major role in the classification of measures in relation to another measure and is particularly used in the Lebesgue decomposition theorem and in the representation theorem in stochastics.

definition

A ( signed or ordinary) measure is called singular with respect to another (signed or ordinary) measure (also singular to or singular ), if there is a set with ${\displaystyle \nu \colon {\mathcal {F}}\rightarrow [0,\infty ]}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle A\in {\mathcal {F}}}$

${\displaystyle \nu (\Omega \setminus A)=0}$and . ${\displaystyle \mu (A)=0}$

The dimensions and are defined in the same measuring room . For " is singular regarding " one writes short . ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle (\Omega ,{\mathcal {F}})}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu \perp \mu }$

Examples

• The zero dimension is singular with respect to any other dimension in any measurement space.
• Each Dirac measure on with respect to the Lebesgue measure singular. ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$
• Every discrete distribution on is singular with respect to the Lebesgue measure. ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$
• The Cantor distribution in the measuring room is a continuous, singular distribution with regard to the Lebesgue measure. ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$
• The following applies to the Hahn-Jordan decomposition of a signed measure . ${\displaystyle \nu =\nu ^{+}-\nu ^{-}}$ ${\displaystyle \nu }$ ${\displaystyle \nu ^{+}\perp \nu ^{-}}$

properties

• The singularity of measures is a symmetrical relation . So it applies

${\displaystyle \nu \perp \mu \Leftrightarrow \mu \perp \nu }$.

• For probability measures with densities, it is true that they are singular to one another if and only if their Hellinger distance is equal to one.

Important statements

The decomposition rate of Lebesgue provides a signed measure and a measure of a decomposition of in a portion which is singular with respect to and in a portion which is absolutely continuous respect is. ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$

literature

• 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 , doi : 10.1007 / 978-3-642-36018-3 .