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A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [[#References|[a1]]]. This theorem was established independently by D.G. Kendall [[#References|[a2]]] and G. Matheron [[#References|[a3]]] in their work on random closed sets.
 
A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [[#References|[a1]]]. This theorem was established independently by D.G. Kendall [[#References|[a2]]] and G. Matheron [[#References|[a3]]] in their work on random closed sets.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102102.png" /> be the family of closed and compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102103.png" />, respectively. The family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102104.png" /> is endowed with the [[Hit-or-miss topology|hit-or-miss topology]], which is generated by
+
Let $  {\mathcal F} $
 +
and $  {\mathcal K} $
 +
be the family of closed and compact subsets of $  \mathbf R  ^ {d} $,  
 +
respectively. The family $  {\mathcal F} $
 +
is endowed with the [[Hit-or-miss topology|hit-or-miss topology]], which is generated by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102105.png" /></td> </tr></table>
+
$$
 +
{\mathcal F} _ {G _ {1}  \dots G _ {n} }  ^ {K} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102106.png" /></td> </tr></table>
+
$$
 +
=  
 +
\left \{ {F \in {\mathcal F} } : {F \cap G _ {i} \neq \emptyset  ( i = 1 \dots n ) ,  F \cap K = \emptyset } \right \} ,
 +
$$
  
for a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102107.png" /> and open sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102108.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c1102109.png" /> be the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021011.png" />-algebra generated by the hit-or-miss topology. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021012.png" /> is the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021013.png" />-algebra of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021014.png" /> containing the sets
+
for a compact set $  K $
 +
and open sets $  G _ {i} $.  
 +
Let $  \Sigma _  {\mathcal F}  $
 +
be the Borel $  \sigma $-
 +
algebra generated by the hit-or-miss topology. Then $  \Sigma _  {\mathcal F}  $
 +
is the smallest $  \sigma $-
 +
algebra of subsets of $  {\mathcal F} $
 +
containing the sets
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021015.png" /></td> </tr></table>
+
$$
 +
{\mathcal F} _ {K} = \left \{ {F \in {\mathcal F} } : {F \cap K \neq \emptyset } \right \} ,  K \in {\mathcal K}.
 +
$$
  
Now, a random closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021016.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021017.png" />-valued random element. Its distribution is described by the corresponding [[Probability measure|probability measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021019.png" />:
+
Now, a random closed set $  \Xi $
 +
is an $  {\mathcal F} $-
 +
valued random element. Its distribution is described by the corresponding [[Probability measure|probability measure]] $  {\mathsf P} $
 +
on $  \Sigma _  {\mathcal F}  $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021020.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} ( {\mathcal F} _ {G _ {1}  \dots G _ {n} }  ^ {K} ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021021.png" /></td> </tr></table>
+
$$
 +
=  
 +
{\mathsf P} ( \Xi \cap K = \emptyset,  \Xi \cap G _ {i} \neq \emptyset,  i = 1 \dots n ) .
 +
$$
  
 
This distribution can also be characterized by the functional
 
This distribution can also be characterized by the functional
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021022.png" /></td> </tr></table>
+
$$
 +
T _  \Xi  ( K ) = {\mathsf P} ( \Xi \cap K \neq \emptyset ) ,  K \in {\mathcal K}.
 +
$$
  
The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021023.png" /> is an alternating Choquet capacity of infinite order. This means that:
+
The functional $  T _  \Xi  $
 +
is an alternating Choquet capacity of infinite order. This means that:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021024.png" /> is upper semi-continuous (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021025.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021026.png" />; cf. also [[Semi-continuous function|Semi-continuous function]]);
+
i) $  T _  \Xi  $
 +
is upper semi-continuous ( $  K _ {n} \downarrow K $
 +
implies $  T _  \Xi  ( K _ {n} ) \downarrow T _  \Xi  ( K ) $;  
 +
cf. also [[Semi-continuous function|Semi-continuous function]]);
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021028.png" />, where
+
ii) $  S _ {n} ( K;K _ {1} \dots K _ {n} ) \geq  0 $,  
 +
$  n \geq  0 $,  
 +
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021029.png" /></td> </tr></table>
+
$$
 +
S _ {0} ( K ) = 1 - T _  \Xi  ( K ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021030.png" /></td> </tr></table>
+
$$
 +
S _ {n} ( K;K _ {1} \dots K _ {n} ) = S _ {n - 1 }  ( K;K _ {1} \dots K _ {n - 1 }  )  -
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021031.png" /></td> </tr></table>
+
$$
 +
-
 +
S _ {n - 1 }  ( K \cup K _ {n} ;K _ {1} \dots K _ {n - 1 }  ) .
 +
$$
  
The values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021032.png" /> can be interpreted as the probability that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021033.png" /> does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021034.png" /> but does intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021035.png" />.
+
The values $  S _ {n} ( K;K _ {1} \dots K _ {n} ) $
 +
can be interpreted as the probability that $  \Xi $
 +
does not intersect $  K $
 +
but does intersect $  K _ {1} \dots K _ {n} $.
  
The Choquet–Kendall–Matheron theorem asserts that given a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021037.png" />, there exists a distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021039.png" /> with
+
The Choquet–Kendall–Matheron theorem asserts that given a functional $  T $
 +
on $  {\mathcal K} $,  
 +
there exists a distribution $  {\mathsf P} $
 +
on $  {\mathcal F} $
 +
with
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021040.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} ( {\mathcal F} _ {K} ) = T ( K ) ,  K \in {\mathcal K},
 +
$$
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021041.png" /> is an alternating Choquet capacity of infinite order with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110210/c11021043.png" />. This distribution is necessarily unique.
+
if and only if $  T $
 +
is an alternating Choquet capacity of infinite order with 0 \leq  T ( K ) \leq  1 $
 +
and $  T ( \emptyset ) = 0 $.  
 +
This distribution is necessarily unique.
  
 
See [[#References|[a4]]] for applications.
 
See [[#References|[a4]]] for applications.

Latest revision as of 16:44, 4 June 2020


A theorem characterizing the distribution of a random closed set in terms of the Choquet capacity functional [a1]. This theorem was established independently by D.G. Kendall [a2] and G. Matheron [a3] in their work on random closed sets.

Let $ {\mathcal F} $ and $ {\mathcal K} $ be the family of closed and compact subsets of $ \mathbf R ^ {d} $, respectively. The family $ {\mathcal F} $ is endowed with the hit-or-miss topology, which is generated by

$$ {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} = $$

$$ = \left \{ {F \in {\mathcal F} } : {F \cap G _ {i} \neq \emptyset ( i = 1 \dots n ) , F \cap K = \emptyset } \right \} , $$

for a compact set $ K $ and open sets $ G _ {i} $. Let $ \Sigma _ {\mathcal F} $ be the Borel $ \sigma $- algebra generated by the hit-or-miss topology. Then $ \Sigma _ {\mathcal F} $ is the smallest $ \sigma $- algebra of subsets of $ {\mathcal F} $ containing the sets

$$ {\mathcal F} _ {K} = \left \{ {F \in {\mathcal F} } : {F \cap K \neq \emptyset } \right \} , K \in {\mathcal K}. $$

Now, a random closed set $ \Xi $ is an $ {\mathcal F} $- valued random element. Its distribution is described by the corresponding probability measure $ {\mathsf P} $ on $ \Sigma _ {\mathcal F} $:

$$ {\mathsf P} ( {\mathcal F} _ {G _ {1} \dots G _ {n} } ^ {K} ) = $$

$$ = {\mathsf P} ( \Xi \cap K = \emptyset, \Xi \cap G _ {i} \neq \emptyset, i = 1 \dots n ) . $$

This distribution can also be characterized by the functional

$$ T _ \Xi ( K ) = {\mathsf P} ( \Xi \cap K \neq \emptyset ) , K \in {\mathcal K}. $$

The functional $ T _ \Xi $ is an alternating Choquet capacity of infinite order. This means that:

i) $ T _ \Xi $ is upper semi-continuous ( $ K _ {n} \downarrow K $ implies $ T _ \Xi ( K _ {n} ) \downarrow T _ \Xi ( K ) $; cf. also Semi-continuous function);

ii) $ S _ {n} ( K;K _ {1} \dots K _ {n} ) \geq 0 $, $ n \geq 0 $, where

$$ S _ {0} ( K ) = 1 - T _ \Xi ( K ) , $$

$$ S _ {n} ( K;K _ {1} \dots K _ {n} ) = S _ {n - 1 } ( K;K _ {1} \dots K _ {n - 1 } ) - $$

$$ - S _ {n - 1 } ( K \cup K _ {n} ;K _ {1} \dots K _ {n - 1 } ) . $$

The values $ S _ {n} ( K;K _ {1} \dots K _ {n} ) $ can be interpreted as the probability that $ \Xi $ does not intersect $ K $ but does intersect $ K _ {1} \dots K _ {n} $.

The Choquet–Kendall–Matheron theorem asserts that given a functional $ T $ on $ {\mathcal K} $, there exists a distribution $ {\mathsf P} $ on $ {\mathcal F} $ with

$$ {\mathsf P} ( {\mathcal F} _ {K} ) = T ( K ) , K \in {\mathcal K}, $$

if and only if $ T $ is an alternating Choquet capacity of infinite order with $ 0 \leq T ( K ) \leq 1 $ and $ T ( \emptyset ) = 0 $. This distribution is necessarily unique.

See [a4] for applications.

References

[a1] G. Choquet, "Theory of capacities" Ann. Inst. Fourier , V (1953-1954) pp. 131–295
[a2] D.G. Kendall, "Foundations of a theory of random sets" E.F. Harding (ed.) D.G. Kendall (ed.) , Stochastic Geometry , Wiley (1974) pp. 322–376
[a3] G. Matheron, "Random sets and integral geometry" , Wiley (1975)
[a4] D. Stoyan, W.S. Kendall, J. Mecke, "Stochastic geometry and its applications" , Wiley (1995) (Edition: Second)
How to Cite This Entry:
Choquet-Kendall-Matheron theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Choquet-Kendall-Matheron_theorem&oldid=19235
This article was adapted from an original article by H.J.A.M. Heijmans (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article