Ionescu-Tulcea theorem


In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem deals with the existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the individual events may be independent or dependent with respect to each other. Thus, the statement goes beyond the mere existence of countable product measures. The theorem was proved by Cassius Ionescu-Tulcea in 1949.

Statement of the theorem

Suppose that is a probability space and for is a sequence of measure spaces. For each let
be the Markov kernel derived from and, where
Then there exists a sequence of probability measures
and there exists a uniquely defined probability measure on, so that
is satisfied for each and.

Applications

The construction used in the proof of the Ionescu-Tulcea theorem is often used in the theory of Markov decision processes, and, in particular, the theory of Markov chains.