Doob Dynkin lemma

The Doob-Dynkin lemma is a statement from probability theory named after the mathematicians Joseph L. Doob and Eugene Dynkin , which establishes a functional relationship between two random variables .

Be and two pictures . In applications there is usually a probability space and and are random quantities defined on it. In probability theory, the question arises when one can already calculate from , that is, when there is a Borel-measurable function such that . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \Omega \rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \Omega }$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle Y=h\circ X}$

Is now a σ-algebra on and is -measurable, it follows as a necessary condition of a measurable function of the existence of that also be -measurable must, because the concatenation of measurable functions is again measured. This condition is strongest when choosing as small as possible, that is, when ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle X}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle Y=h\circ X}$ ${\displaystyle Y}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle {\mathcal {A}}=\sigma (X):=\{X^{-1}(B);\,B\subset \mathbb {R} ^{n}\,{\rm {Borelmenge}}\}}$,

is the so-called σ-algebra generated by . That this condition is then even sufficient means precisely that ${\displaystyle X}$

Doob-Dynkin lemma : The following statements are equivalent for two mappings : ${\displaystyle X,Y:\Omega \rightarrow \mathbb {R} ^{n}}$

1. There is a Borel measurable function with . ${\displaystyle h:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle Y=h\circ X}$
2. ${\displaystyle Y}$is measurable. ${\displaystyle \sigma (X)}$

This makes it understandable that σ-algebras are viewed as carriers of probabilistic information. If the σ-algebra generated by is measurable, it can not contain any information that is not already contained in , as specified by the first statement. ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X}$

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• A. Bobrowski: Functional analysis for probability and stochastic processes: an introduction , Cambridge University Press (2005), ISBN 0-521-83166-0
• MM Rao, RJ Swift: Probability Theory with Applications , Mathematics and Its Applications, Volume 582, Springer-Verlag (2006), ISBN 0-387-27730-7