Reading about random sets, I've come across the phrase " B ( F ) is...

Santino Bautista

Santino Bautista

Answered question


Reading about random sets, I've come across the phrase " B ( F ) is the Borel σ-algebra generated by the topology of closed convergence." where F is the family of closed sets in R .
I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel σ-algebra on a set S is the σ-algebra generated by the open sets of S, and I suppose a topology gives a notion of "openness", but I would appreciate if someone could provide me some intuition behind this particular σ-algebra and the "topology of closed convergence".

Answer & Explanation

Patricia Curry

Patricia Curry

Beginner2022-06-22Added 15 answers

For any family F of subsets of a set X we can talk about the σ-algebra generated by F on X, which is the defined to be the intersection of all σ-algebras on X that contain F as a subset. This is standard in measure theory. It's well-defined as the intersection of σ-algebras is a σ-algebra again and we always have at least one σ-algebra containing F , namely the power set of X.
A topology on X is just another such family F so the definition applies. The σ-algebra will thus contain all open sets, all closed sets (their complements), all G δ sets and F σ sets etc. (all in that topology). It's also called the Borel σ-algebra of a topological space.

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