Extension set by Kolmogorov

The Kolmogorov extension theorem , sometimes Kolmogorov'scher extension kit , set of Kolmogorov or existence theorem of Kolmogorov called, is a central existence theorem of probability theory . The statement is ascribed to Andrei Nikolajewitsch Kolmogorow , but also called the theorem of Daniell-Kolmogorov , since it was already proven in 1919 by Percy John Daniell in a non-stochastic formulation.

The theorem provides the existence of probability measures on uncountable product spaces and is therefore essential for the existence of stochastic processes , countable and uncountable product measures and independently identically distributed random variables .

statement

Given a non-empty index set and Borel spaces for . Let be the set of all non-empty, finite subsets of . If a projective family of probability measures is given, then there is a clearly defined probability measure in the measurement space ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {E}}(I)}$ ${\displaystyle I}$ ${\displaystyle (P_{J})_{J\in {\mathcal {E}}(I)}}$ ${\displaystyle P}$

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

for which applies to everyone . The projection on the components of the index set denotes . Then you write ${\displaystyle P_{J}=P\circ (\pi _{J}^{I})^{-1}}$ ${\displaystyle J\in {\mathcal {E}}(I)}$ ${\displaystyle \pi _{J}^{I}}$ ${\displaystyle J}$

${\displaystyle \varprojlim _{J\uparrow I}P_{J}=:P}$

and then designates the probability measure as the projective limit . ${\displaystyle P}$

Example: Product dimensions on uncountable products

If one considers an uncountable index set as well as Borel spaces , each provided with a probability measure for all , then the product measure for anything can be found on finite products ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle P_{i}}$ ${\displaystyle i\in I}$ ${\displaystyle J\in {\mathcal {E}}(I)}$

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

construct in the conventional way of mass theory. The family of product measures but projective and thus be a unique probability measure after the above sentence on ${\displaystyle (P_{J})_{J\in {\mathcal {E}}(I)}}$ ${\displaystyle P}$

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

continue. The set of Andersen Jessen delivers a more general statement on the existence of any product dimensions, can be dispensed with in the use of Borel spaces.

See also

• Projective Limes

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 295.
2. ^ Schmidt: Measure and Probability. 2011, p. 458.
3. ↑ Meintrup, Schäffler: Stochastics. 2005, p. 559.
4. ^ "But you have to remember PJ Daniell of Sheffield" - John Aldrich. Electronic Journal for History of Probability and Statistics website. Retrieved November 7, 2015.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 294-296 , doi : 10.1007 / 978-3-642-36018-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 558-561 , doi : 10.1007 / b137972 .
• Kaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 458-461 , doi : 10.1007 / 978-3-642-21026-6 .