A continuity theorem for cores of random closed sets

P. Terán (2008). Proceedings of the American Mathematical Society. 136, 4417-4425.


The starting point is Zvi Artstein's 1983 paper on distributions of random sets. Among other things, he proved that, if a sequence of distributions of random compact sets converges weakly in the Hausdorff metric, their cores (or sets of distributions of selections) also converge in the Hausdorff metric defined in the space of compact sets of distributions. The proof is a quite laborious one, and I have never been able to read it through.

On a rainy summer afternoon I killed some time by proving it using the Skorokhod representation theorem. I worked a bit harder and adapted the new proof to get an extension to the unbounded case in locally compact separable Hausdorff spaces with the Fell topology. Then I checked Artstein again and saw that he already knew that (using the one-point compactification-- quite smarter than me). I realized that, to get something publishable, I would need a proof of the general unbounded case.

By Christmas, I had that general proof. It involves some results and notions from Hyperspace Topology which were developed in the nineties. It is a quite symphonic proof, with a large number of elements assembled together in a very nice way.


Up the line:
This starts a new line.

Down the line:
Nothing yet. I have some material I will finish and prepare for publication as soon as 36-hour days are available.

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