Sunday, 14 April 2013

Distributions of random closed sets via containment functionals

P. Terán (2014). Journal of Nonlinear and Convex Analysis 15, 907-917.

A central problem in the theory of random sets is how to characterize the distribution of a random set in a simpler way. The fact that we are dealing with a random element of a space each point of which is a set implies that the distribution is defined on a sigma-algebra which is a set of sets of sets.

The standard road, initiated in the seventies by e.g. Kendall and Matheron (but already travelled in the opposite direction in the fifties by Choquet) is to describe a set by a number of 0-1 characteristics, typically whether it hits (i.e. intersects) or not each element of a family of test sets. This gives us the hitting functional defined on the test sets (a set of sets, one order of magnitude simpler) as the hitting probability of the random set.

The classical assumptions on the underlying space are: locally compact, second countable and Hausdorff (this implies the existence of a separable complete metric). That is enough for applications in Rd but seems insufficiently general to live merrily ever after. In contrast, the theory of probability measures in metric spaces was well developed without local compactness about half a century ago.

Molchanov's book includes three proofs of the Choquet-Kendall-Matheron theorem, and it is fascinating how all three break down in totally different ways if local compactness is dropped.

This paper is an attempt at finding a new path of proof that avoids local compactness. I failed but ended up succeeding in replacing second countability by sigma-compactness, which (under local compactness) is strictly weaker. Sadly, I didn't know how to handle some problems and had to opt for sigma-compactness after believing for some time that I had a correct proof in locally compact Hausdorff spaces.

Regarding the assumption I initially set out to defeat, all I can say for the moment is that now there are four proofs that break down in non-locally-compact spaces.

Up the line:
This starts a new line.

Down the line:
Some work awaits its moment to be typed.

To download, click on the title or here.

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