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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.

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

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

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

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|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|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 [[#References|[a4]]] for applications.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Choquet,   "Theory of capacities"  ''Ann. Inst. Fourier'' , '''V'''  (1953-1954)  pp. 131–295</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Matheron,   "Random sets and integral geometry" , Wiley  (1975)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Stoyan,   W.S. Kendall,   J. Mecke,   "Stochastic geometry and its applications" , Wiley  (1995)  (Edition: Second)</TD></TR></table>