Andersen-Jessen theorem

The set of Andersen Jessen is a mathematical theorem from probability theory , which deals with the existence of product dimensions of any number of probability measures employed. In contrast to many other statements of existence such as Ionescu-Tulcea's theorem and Kolmogorov's extension theorem , he also allows products from an uncountable number of measuring spaces and makes no further demands on the structure of these spaces. The theorem thus provides, for example, the existence of uncountable families of independent random variables and of uncountable product models. The set is named after the Danish mathematicians Erik Sparre Andersen and Børge Jessen .

statement

A non-empty index set as well as measurement spaces and probability measures defined on them are given for all . Denote with the set of all finite subsets of and for the product of the measuring spaces as ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle P_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {E}}(I)}$ ${\displaystyle I}$ ${\displaystyle K\subset I}$

${\displaystyle (\Omega ^{K},{\mathcal {A}}^{K}):=\left(\prod _{i\in K}\Omega _{i},\bigotimes _{i\in K}A_{i}\right)}$.

Furthermore, be

${\displaystyle \pi _{J}:\Omega ^{I}\to \Omega ^{J}\;{\text{ definiert durch }}\pi (\omega )=\omega |_{J}}$

the projection onto the components and the image measure of a probability measure under the projection onto the components ( the distribution of is equivalent ). ${\displaystyle J}$ ${\displaystyle Q_{\pi _{J}}:=Q\circ (\pi _{J})^{-1}}$ ${\displaystyle Q}$ ${\displaystyle Q_{\pi _{J}}}$ ${\displaystyle \pi _{J}}$

The theorem now says that there is exactly one probability measure on , so that for all always ${\displaystyle P}$ ${\displaystyle (\Omega ^{I},{\mathcal {A}}^{I})}$ ${\displaystyle J\in {\mathcal {E}}(I)}$

${\displaystyle P_{\pi _{J}}=\bigotimes _{i\in J}P_{i}}$

applies. The projections onto a finite number of components therefore always agree with the finite product dimension.

history

Statements on the existence of infinite product measures were proven early on, and this is where measure theory and probability theory differ from one another. One of the central theorems on this topic is Kolmogorov's extension theorem , which, however, was previously proven by Percy John Daniell in a non-stochastic formulation. However, this theorem provides the existence of a probability measure only under the precondition that the measuring spaces involved have a certain structure; they must be Borelian . For this, the projections of the probability measures can also be dependent on one another, which makes this theorem particularly interesting for the theory of stochastic processes.

If one only demands that the individual probability measures are independent, then the existence of a measure on the product space can be shown for any measurement spaces. The first work on this topic can be traced back to Antoni Łomnicki and Stanisław Marcin Ulam and John von Neumann .

A natural question is now whether the two statements can be combined, i.e. whether there is a measure on the product space whose projections onto the components are not necessarily independent and without requiring additional structure on the measuring spaces. The achievement of Erik Sparre Andersen and Børge Jessen is that in 1948 they showed with a counterexample that this is impossible.

Individual evidence

1. ^ Schmidt: Measure and Probability. 2011, p. 210.
2. ^ Børge Jessen - Christian Berg website of the Electronic Journal for History of Probability and Statistics. Retrieved November 15, 2015
3. ↑ On the introduction of product measures on infinite sets - Erik Sparre Andersen, Børge Jessen ( Memento of the original from November 17, 2015 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. Online version of the 1948 publication on the website of the Syddansk Universitet . Retrieved November 15, 2015 

literature

• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 209-217 , doi : 10.1007 / 978-3-642-21026-6 .