# Choquet-Kendall-Matheron theorem

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=46339
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