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Random closed set

Type of random variable

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Source updated September 26, 2025

In mathematics, particularly in probability theory and stochastic geometry, a random closed set is a random variable whose values are closed subsets of a given topological space, typically Euclidean space $\mathbb {R} ^{n}$ . Random closed sets generalize the concept of random variables and random processes by allowing entire sets, rather than individual points or vectors, to be treated as random elements. They are widely used in areas such as spatial statistics, image analysis, materials science, and mathematical morphology.

01Definition

A random closed set in $\mathbb {R} ^{d}$ is a measurable function from a probability space $(\Omega ,{\mathcal {A}},P)$ into $({\mathcal {F}},\Sigma )$ . Here ${\mathcal {F}}$ is the collection of all closed subsets of $\mathbb {R} ^{d}$ and $\Sigma$ is the sigma-algebra generated over ${\mathcal {F}}$ by the sets ${\mathcal {F}}_{K}=\{F\in {\mathcal {F}}:F\cap K=\emptyset \}$ for all compact subsets $K\subset \mathbb {R} ^{d}$ .

02History

Mentions of random sets have appeared for almost a century beginning with A.N. Kolmogorov's book, Foundations of the Theory of Probability, which provided the axiomatic foundation for probability theory. In this book, Kolmogorov defined what is now referred to as a random set. Up until the 1960s, mentions of random sets could be found scattered throughout publications before Gustave Choquet formalized the concept of a random set. French mathematician Georges Matheron is recognized as the first person to concentrate on random sets with closed values and formulate a definition.

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Sources and credits

This article is adapted from the Wikipedia article “Random closed set”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Random closed set

01Definition

02History

Sources and credits

Related topics

Random compact set