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

====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|Stochastic process]]; [[Joint distribution|Joint distribution]]).