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)