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<p>We study the question whether the event that a random closed set in a metric space is compact, is measurable. If not, the probability that the realizations of a random closed set are compact would turn out to be undefined. Among the results, in every metrizable Suslin space whose topology does not come from a complete metric, there are random sets for which the answer is negative. Two ways to overcome this difficulty are proposed. If the random closed set is Borel measurable with respect to the Hausdorff metric, then the answer is always positive. Alternatively, several sufficient conditions on the space are given under which that event can only differ from a measurable event by a null set, whence its probability is meaningfully defined in the completion of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>-algebra.</p>

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On the Measurability of the Event that a Random Closed Set is Compact

  • 期刊论文

  • 发表日期：

    2026年2月2日

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  • Pedro Terán

摘要

We study the question whether the event that a random closed set in a metric space is compact, is measurable. If not, the probability that the realizations of a random closed set are compact would turn out to be undefined. Among the results, in every metrizable Suslin space whose topology does not come from a complete metric, there are random sets for which the answer is negative. Two ways to overcome this difficulty are proposed. If the random closed set is Borel measurable with respect to the Hausdorff metric, then the answer is always positive. Alternatively, several sufficient conditions on the space are given under which that event can only differ from a measurable event by a null set, whence its probability is meaningfully defined in the completion of the \(\sigma \) σ -algebra.