View source for Choquet-Kendall-Matheron theorem

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

<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>

<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>

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

<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>

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" />:

<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>

<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>

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>

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:

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]]);

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

<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>

<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>

<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>

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 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

<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>

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.

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>