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''projective system of probability measures, consistent system of probability measures, consistent system of 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|Measure]]. A more general construction is given below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236701.png" /> be an index set with a pre-order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236702.png" /> filtering to the right; suppose one is given a projective system of sets: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236703.png" /> there is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236704.png" /> and for every pair of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236705.png" /> there is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236706.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236707.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236708.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c0236709.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367010.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367011.png" /> be the identity mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367012.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367013.png" />. It is further assumed that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367014.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367015.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367016.png" /> of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367017.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367018.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367020.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367021.png" /> is measurable. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367022.png" /> be a given distribution (or, more generally, a measure) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367023.png" />, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367024.png" />. The system of distributions (measures) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367025.png" /> is called compatible (or consistent, or a projective system of distributions (measures)) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367026.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367027.png" />. Under certain additional conditions on the projective limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367028.png" />, there is a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367029.png" /> (the projective limit of the projective system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367030.png" />) such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367031.png" /> is the canonical projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367032.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367035.png" />.
+
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====
 
====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>
 
<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====
 
====Comments====
A partial order or pre-order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367037.png" /> is said to filter to the right if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367038.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367041.png" />. The projective limit measure exists if, for instance, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367042.png" /> are all compact spaces, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367043.png" /> are all surjective and the family of norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367044.png" /> is bounded, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367047.png" /> continuous of compact support. It also exists if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367048.png" /> are compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367049.png" /> surjective, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367050.png" /> are positive measures; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367051.png" /> is positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367052.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023670/c02367053.png" />.
+
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]]).
 
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]]).

Latest revision as of 17:45, 4 June 2020


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).

How to Cite This Entry:
Compatible distributions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Compatible_distributions&oldid=46415
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article