Doob–Dynkin lemma

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

Let and be two pictures. In applications is usually a probability space and random variables and are defined thereon. In probability theory, the question arises as to when you can already calculate from, that is, when there is a Borel measurable function such that.

Is now a σ - algebra and is - measurable, it follows with a necessary condition for the existence of a measurable function that well - measurable must be, because the concatenation of measurable functions is again measurable. This condition is most severe when one selects as low as possible, that is, when

,

Is the so-called σ - algebra generated by. The fact that this condition is sufficient even then, just stating the

Doob - Dynkin Lemma: The following statements are equivalent for two pictures:

Thus it is understandable that one σ - algebras sees as carrier of probabilistic information. With respect to the σ - algebra generated by measurable, so can not contain information that is not already plugged in, as clarified by the first statement.

Swell

  • 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
  • Set ( mathematics)
  • Stochastics
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