Projective family of probability measures

A projective family of probability measures , or projective family for short , sometimes also called a consistent family (of probability measures) , is in probability theory a family of probability measures whose distributions of the projections on the components are subject to special requirements. Projective families are used, for example, in the proof of Andersen-Jessen's theorem or the formulation of Kolmogorov's extension theorem , which guarantees the existence of probability measures with predetermined properties on uncountable product spaces and thereby also provides important statements about the existence of stochastic processes .

definition

Given any non-empty index set and measurement spaces for . For any was ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in I}$ ${\displaystyle K\subset I}$

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

the product of the measuring rooms and

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

the projection onto the components of the index set . Furthermore, let the set of all non-empty, finite subsets of . ${\displaystyle L\subset J\subset I}$ ${\displaystyle {\mathcal {E}}(I)}$ ${\displaystyle I}$

A family of probability measures is called a projective family of probability measures if, for every subset of the finite set , that ${\displaystyle (P_{J})_{J\in {\mathcal {E}}(I)}}$ ${\displaystyle L\subset J}$ ${\displaystyle J}$

${\displaystyle P_{L}=P_{J}\circ (\pi _{L}^{J})^{-1}}$

is. The probability measures of the smaller index set should therefore agree with the distribution of the probability measures of the large index set under the projection onto the components.

example

An arbitrary index set and a measurement space are given ${\displaystyle I}$

${\displaystyle \left(\prod _{i\in I}\Omega _{i},\bigotimes _{i\in I}{\mathcal {A}}_{i}\right)}$

provided with a probability measure . Due to the properties of the projection applies to . So every family is ${\displaystyle P}$ ${\displaystyle \pi _{L}^{I}=\pi _{J}^{I}\circ \pi _{L}^{J}}$ ${\displaystyle I\supset J\supset L}$

${\displaystyle (P_{J}:=P\circ (\pi _{J}^{I})^{-1})_{J\in {\mathcal {E}}(I)}}$

projective.

comment

The above example shows that the projectivity of a family of probability measures is necessary for the existence of a probability measure on the product space. For Borelian spaces , Kolmogorov's extension theorem also provides the converse. Here the projective family already clearly determines a probability measure on the product space.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 294 , doi : 10.1007 / 978-3-642-36018-3 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 204-208 , doi : 10.1007 / 978-3-642-21026-6 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6 , pp. 555 , doi : 10.1007 / b137972 .