Measure of uniqueness

The uniqueness of measure theorem , also simply called the uniqueness theorem within the relevant context , is a mathematical statement from the mathematical sub-areas of measure theory and stochastics . He deals with the question of when an abstract concept of volume, i.e. a measure or, more specifically, a probability measure, has already been uniquely determined.

Some more specific sentences, such as the correspondence sentence, can be derived directly from the measure of uniqueness . The structural implications of the measure uniqueness theorem are just as important, since they have a decisive influence on which set systems are suitable for the construction of measures if they are to be clearly determined.

statement

Depending on the area of ​​application, the sentence is worded slightly differently. In the measure theory, the more general version for σ-finite measures is listed, in the probability theory mostly the special case for probability measures .

Dimension theory version

Let a set and a σ-algebra with producer be given . So it applies ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle {\mathcal {A}}=\sigma ({\mathcal {E}})}$.

Furthermore, two dimensions and are given. Then: ${\displaystyle \mu }$ ${\displaystyle \nu }$ ${\displaystyle {\mathcal {A}}}$

Is average stable , amounts exist in so ${\displaystyle {\mathcal {E}}}$ ${\displaystyle E_{1},E_{2},E_{3},\dots }$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle X=\bigcup _{i=1}^{\infty }E_{i}}$

and is

${\displaystyle \mu (E)=\nu (E)}$ for all ${\displaystyle E\in {\mathcal {E}}}$

such as

${\displaystyle \mu (E_{n})=\nu (E_{n})<\infty }$for all , ${\displaystyle n\in \mathbb {N} }$

so is . ${\displaystyle \mu =\nu }$

Probability Theory Version

Let a set and a σ-algebra with producer be given . So it applies ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {E}}}$

${\displaystyle {\mathcal {A}}=\sigma ({\mathcal {E}})}$.

Furthermore, two probability measures and are given. Then: ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle {\mathcal {A}}}$

Is average stable and always applies to everyone ${\displaystyle {\mathcal {E}}}$ ${\displaystyle E\in {\mathcal {E}}}$

${\displaystyle P(E)=Q(E)}$,

so is . ${\displaystyle P=Q}$

Implications

One implication of the uniqueness theorem is to choose set systems that are stable in terms of the definition when defining set functions such as content and premeasures . This guarantees that if the set function can be continued for a measure on a corresponding σ-algebra containing the set system, this continuation is also unique. Typical examples of such set systems are half rings .

Two further conclusions from the uniqueness theorem are the uniqueness of the (finite) product measure and the correspondence theorem, which is important for stochastics and which illuminates the relationship between probability measures and distribution functions . ${\displaystyle \mathbb {R} }$

Evidence sketch

The probabilistic version can be shown according to the principle of proof of the good sets as follows: First we consider the set system

${\displaystyle {\mathcal {D}}:=\{A\in {\mathcal {A}}\mid P(A)=Q(A)\}}$

those sets on which the probability measures agree. This system of quantities is a Dynkin system because

• the stability with regard to countably many disjoint unions follows from the σ-additivity of the probability measures
• the amount is included because it always applies ${\displaystyle \Omega }$ ${\displaystyle P(\Omega )=Q(\Omega )=1}$
• the stability with regard to the formation of the difference follows from the subtractivity of the probability measures.

According to the prerequisite

${\displaystyle {\mathcal {D}}\supset {\mathcal {E}}}$.

If you now consider the Dynkin system generated by ${\displaystyle {\mathcal {E}}}$ , then the following applies due to its minimalism ${\displaystyle \delta ({\mathcal {E}})}$

${\displaystyle {\mathcal {D}}\supset \delta ({\mathcal {E}})}$

But since it is cut stable, according to Dynkin's π-λ theorem applies ${\displaystyle {\mathcal {E}}}$

${\displaystyle \delta ({\mathcal {E}})=\sigma ({\mathcal {E}})}$.

So is

${\displaystyle {\mathcal {D}}\supset \delta ({\mathcal {E}})=\sigma ({\mathcal {E}})={\mathcal {A}}}$.

At the same time, however, by definition, always applies ${\displaystyle {\mathcal {D}}}$

${\displaystyle {\mathcal {D}}\subset {\mathcal {A}}}$,

what then because of and immediately ${\displaystyle {\mathcal {D}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {D}}\supset {\mathcal {A}}}$

${\displaystyle {\mathcal {D}}={\mathcal {A}}}$

follows. The two probability measures therefore agree on the entire σ-algebra.

The proof of the measure-theoretic version follows essentially the same idea, but uses an exhaustion argument in combination with the σ-continuity of the measures to show the agreement on all sets.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 60-61 , 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 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , doi : 10.1007 / b137972 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 63-64 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 23.
2. ^ Georgii: Stochastics. 2009, p. 16.
3. ↑ Klenke: Probability Theory. 2013, pp. 19-20.