Lebesgue decomposition (functions)

This article deals with the Lebesgue decomposition of real functions. For the Lebesgue decomposition of (signed) dimensions, see Lebesgue decomposition theorem .

The Lebesgue decomposition of a real function is a measure theoretical statement that divides a function into three functions with clearly defined properties. A special case of this is the representation theorem from stochastics . He dismantled probability measures on about the Lebesgue decomposition of the distribution function uniquely in an absolutely continuous , a discrete and a continuous singular part. ${\displaystyle \mathbb {R} }$

The statement was shown by Henri Léon Lebesgue in 1904.

statement

It is the Lebesgue-Borel measure . Given is a monotonically growing , right-hand continuous function ${\displaystyle \beta }$

${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$.

Then - is differentiable almost everywhere and it denotes the derivative which is defined almost everywhere. ${\displaystyle F}$ ${\displaystyle \beta }$ ${\displaystyle F'}$ ${\displaystyle \beta }$

Then the following applies: there is a clear decomposition

${\displaystyle F=A+S+D}$,

so that

• ${\displaystyle A(0)=0}$is and is a monotonically increasing absolutely continuous function . ${\displaystyle A}$
• ${\displaystyle S(0)=0}$and is a monotonically increasing singular function . ${\displaystyle S}$
• ${\displaystyle D}$a monotonically increasing, right continuous step function is

For the associated Lebesgue-Stieltjes measures or then applies ${\displaystyle \mu _{F}}$ ${\displaystyle \mu _{A},\mu _{S},\mu _{D}}$

${\displaystyle \mu _{F}=\mu _{A}+\mu _{S}+\mu _{D}}$.

The following also applies:

• ${\displaystyle \mu _{D}}$is the purely atomic part of ${\displaystyle \mu _{F}}$
• ${\displaystyle \mu _{A}+\mu _{S}}$is the atomless part of . ${\displaystyle \mu _{F}}$
• ${\displaystyle \mu _{A}}$is absolutely continuous with respect to the Lebesgue-Borel measure and has the Radon-Nikodym density with respect to the Lebesgue-Borel measure. So it applies to measurable ${\displaystyle \beta }$ ${\displaystyle F'}$ ${\displaystyle M}$

${\displaystyle \mu _{A}(M)=\int _{M}F'\mathrm {d} \beta }$.

• ${\displaystyle \mu _{S}}$is singular with respect to the Borel measure.

Representation set

The notation theorem follows directly from the Lebesgue decomposition. The normalization conditions are dropped because distribution functions in the sense of stochastics are already defined via the conditions and . The statement then reads: ${\displaystyle A(0)=S(0)=0}$ ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ ${\displaystyle \lim _{x\to \infty }F(x)=1}$

Given a probability measure on with distribution function . Then there are clearly definite numbers with such that ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle F}$ ${\displaystyle a_{1},a_{2},a_{3}\geq 0}$ ${\displaystyle a_{1}+a_{2}+a_{3}=1}$

${\displaystyle F=a_{1}A+a_{2}S+a_{3}D}$.

Here is

• ${\displaystyle A}$the distribution function of an absolutely continuous probability distribution
• ${\displaystyle S}$the distribution function of a continuously singular probability distribution and
• ${\displaystyle D}$the distribution function of a discrete probability distribution .

Each probability distribution can therefore be split into a continuous, a discrete and a continuously singular part.

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 ).

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 .
• 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 .

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 308.
2. ^ Schmidt: Measure and probability. 2011, p. 262.