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 ( Ω 0 , A 0 , P 0 ) {\displaystyle (\Omega _{0},{\mathcal {A}}_{0},P_{0})} is a probability space and ( Ω i , A i ) {\displaystyle (\Omega _{i},{\mathcal {A}}_{i})} for i ∈ N {\displaystyle i\in \mathbb {N} } is a sequence of measurable spaces. For each i ∈ N {\displaystyle i\in \mathbb {N} } let
κ i : ( Ω i − 1 , A i − 1 ) → ( Ω i , A i ) {\displaystyle \kappa _{i}\colon (\Omega ^{i-1},{\mathcal {A}}^{i-1})\to (\Omega _{i},{\mathcal {A}}_{i})}
be the Markov kernel derived from ( Ω i − 1 , A i − 1 ) {\displaystyle (\Omega ^{i-1},{\mathcal {A}}^{i-1})} and ( Ω i , A i ) , {\displaystyle (\Omega _{i},{\mathcal {A}}_{i}),} , where
Ω i := ∏ k = 0 i Ω k and A i := ⨂ k = 0 i A k . {\displaystyle \Omega ^{i}:=\prod _{k=0}^{i}\Omega _{k}{\text{ and }}{\mathcal {A}}^{i}:=\bigotimes _{k=0}^{i}{\mathcal {A}}_{k}.}
Then there exists a sequence of probability measures
P i := P 0 ⊗ ⨂ k = 1 i κ k {\displaystyle P_{i}:=P_{0}\otimes \bigotimes _{k=1}^{i}\kappa _{k}} defined on the product space for the sequence ( Ω i , A i ) {\displaystyle (\Omega ^{i},{\mathcal {A}}^{i})} , i ∈ N , {\displaystyle i\in \mathbb {N} ,}
and there exists a uniquely defined probability measure P {\displaystyle P} on ( ∏ k = 0 ∞ Ω k , ⨂ k = 0 ∞ A k ) {\displaystyle \left(\prod _{k=0}^{\infty }\Omega _{k},\bigotimes _{k=0}^{\infty }{\mathcal {A}}_{k}\right)} , so that
P i ( A ) = P ( A × ∏ k = i + 1 ∞ Ω k ) {\displaystyle P_{i}(A)=P\left(A\times \prod _{k=i+1}^{\infty }\Omega _{k}\right)}
is satisfied for each A ∈ A i {\displaystyle A\in {\mathcal {A}}^{i}} and i ∈ N {\displaystyle i\in \mathbb {N} } . (The measure P {\displaystyle P} has conditional probabilities equal to the stochastic kernels.)
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.
See also Disintegration theorem Regular conditional probability
Sources Klenke, Achim (2013). Wahrscheinlichkeitstheorie (3rd ed.). Berlin Heidelberg: Springer-Verlag. pp. 292–294. doi:10.1007/978-3-642-36018-3. ISBN 978-3-642-36017-6. Kusolitsch, Norbert (2014). Maß- und Wahrscheinlichkeitstheorie: Eine Einführung (2nd ed.). Berlin; Heidelberg: Springer-Verlag. pp. 169–171. doi:10.1007/978-3-642-45387-8. ISBN 978-3-642-45386-1.
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