Compatible distributions

projective system of probability measures, consistent system of probability measures, consistent system of distributions

A concept in probability theory and measure theory. For the most common and most important case of a product of spaces, see the article Measure. A more general construction is given below. Let $ I $ be an index set with a pre-order relation $ \leq $ filtering to the right; suppose one is given a projective system of sets: For every $ i \in I $ there is a set $ X _ {i} $ and for every pair of indices $ i \leq j $ there is a mapping $ \pi _ {ij} $ of $ X _ {j} $ into $ X _ {i} $ such that $ \pi _ {ik} = \pi _ {ij} \circ \pi _ {jk} $ for $ i \leq j \leq k $; let $ \pi _ {ii} $ be the identity mapping on $ X _ {i} $ for every $ i \in I $. It is further assumed that for each $ i \in I $ there is a $ \sigma $- algebra $ S _ {i} $ of subsets of $ X _ {i} $ such that for $ i \leq j $ the mapping $ \pi _ {ij} $ of $ ( X _ {j} , S _ {j} ) $ into $ ( X _ {i} , S _ {i} ) $ is measurable. Finally, let $ \mu _ {i} $ be a given distribution (or, more generally, a measure) on $ S _ {i} $, for every $ i \in I $. The system of distributions (measures) $ \{ \mu _ {i} \} $ is called compatible (or consistent, or a projective system of distributions (measures)) if $ \mu _ {i} = \mu _ {j} \pi _ {ij} ^ {-} 1 $ whenever $ i \leq j $. Under certain additional conditions on the projective limit $ X = \lim\limits _ \leftarrow ( X _ {i} , \pi _ {ij} ) $, there is a measure $ \mu $( the projective limit of the projective system $ \{ \mu _ {i} \} $) such that if $ \pi _ {i} $ is the canonical projection of $ X $ to $ X _ {i} $, then $ \mu _ {i} = \mu \pi _ {i} ^ {-} 1 $ for all $ i \in I $.

References

[1]	A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2]	S. Bochner, "Harmonic analysis and the theory of probability" , Univ. California Press (1955)
[3]	M. Metivier, "Limites projectives de measures. Martingales. Applications" Ann. Mat. Pura Appl. , 63 (1963) pp. 225–352
[4]	N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French)

Comments

A partial order or pre-order relation $ \leq $ on $ I $ is said to filter to the right if for every $ i , j \in I $ there is a $ k \in I $ such that $ i \leq k $ and $ j \leq k $. The projective limit measure exists if, for instance, the $ X _ {i} $ are all compact spaces, the $ \pi _ {ij} $ are all surjective and the family of norms $ \| \mu _ {i} \| $ is bounded, where $ \| \mu _ {i} \| = \inf \{ {M } : {| \mu _ {i} ( f ) | \leq M \| f \| } \} $, $ \| f \| = \sup _ {x} | f ( x) | $, $ f $ continuous of compact support. It also exists if the $ X _ {i} $ are compact, $ \pi _ {ij} $ surjective, and the $ \mu _ {i} $ are positive measures; then $ \mu $ is positive and $ \| \mu \| = \| \mu _ {i} \| $ for all $ i $.

The concept of consistency (compatibility) of distributions (or measures) is of special importance in the construction of stochastic processes (cf. Stochastic process; Joint distribution).