Lebesgue decomposition theorem

This article covers the Lebesgue decomposition of (signed) measures. For the Lebesgue decomposition of functions see Lebesgue decomposition (functions) .

The decomposition theorem of Lebesgue , even Lebesguescher decomposition theorem called, is a mathematical theorem of the measure theory , a branch of mathematics that deals with the properties of generalized volume terms. It provides the existence and uniqueness of a division of a signed measure into a singular signed measure and an absolutely continuous signed measure with respect to a given measure. This decomposition is then also called the Lebesgue decomposition .

The decomposition theorem of Lebesgue in 1910 by Henri Léon Lebesgue for the Lebesgue measure on proven. A first generalization on Lebesgue-Stieltjes measures comes from Johann Radon , the general proof was provided by Hans Hahn . ${\displaystyle \mathbb {R} ^{n}}$

motivation

With a quasi-integrable function , one can pass through on a dimension space ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle f}$

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

a signed measure to define. The function is then used as density with respect to designated. is then absolutely continuous with respect to , that is, every -null set is also a -null set. ${\displaystyle \nu }$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle f}$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \nu }$

Every signed measure with a density with respect to is consequently absolutely continuous with respect to . The Radon-Nikodym theorem provides the reverse: If a signed measure absolutely continuous with respect , then there exists a density function , then the signed measure that how can show up. ${\displaystyle f}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle f}$

This question can now be expanded: Can , under the assumption that is not absolutely continuous with respect to , be broken down into an absolutely continuous part and a "singular" part ? So exist signed extent with so continuous with respect to absolute and singular respect is? Lebesgue's decomposition theorem answers this question positively. ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \mu }$ ${\displaystyle \nu _{a}}$ ${\displaystyle \nu _{s}}$ ${\displaystyle \nu _{a},\nu _{s}}$ ${\displaystyle \nu =\nu _{a}+\nu _{s}}$ ${\displaystyle \nu _{a}}$ ${\displaystyle \mu }$ ${\displaystyle \nu _{s}}$ ${\displaystyle \mu }$

statement

A measurement space and a σ-finite measure and a σ-finite signed measure on this measurement space are given. Then there is a clear decomposition ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \mu }$ ${\displaystyle \nu }$

${\displaystyle \nu =\nu _{a}+\nu _{s}}$

in two σ-finite signed measures such that ${\displaystyle \nu _{a},\nu _{s}}$

• ${\displaystyle \nu _{a}\ll \mu }$is. is therefore absolutely continuous with respect to ${\displaystyle \nu _{a}}$ ${\displaystyle \mu }$
• ${\displaystyle \nu _{s}\perp \mu }$is. and are therefore singular to one another . ${\displaystyle \nu _{s}}$ ${\displaystyle \mu }$

The signed dimensions are finite exactly when is finite. The decomposition theorem also applies if there is a σ-finite measure, then there are also measures. ${\displaystyle \nu _{a},\nu _{s}}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \nu _{a},\nu _{s}}$

Web links

• VV Sazonov: Lebesgue decomposition . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Lebesgue Decomposition . In: MathWorld (English).

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 286.

literature

• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .
• 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 .