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.
[1][2] Suppose that
is a probability space and
is a sequence of measurable spaces.
let be the Markov kernel derived from
, where Then there exists a sequence of probability measures and there exists a uniquely defined probability measure
, so that is satisfied for each
(The measure
has conditional probabilities equal to the stochastic kernels.
)[3] 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.