Santino Bautista

2022-06-21

Reading about random sets, I've come across the phrase "$\mathcal{B}(\mathcal{F})$ is the Borel $\sigma $-algebra generated by the topology of closed convergence." where $\mathcal{F}$ is the family of closed sets in $\mathbb{R}.$

I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel $\sigma $-algebra on a set $S$ is the $\sigma $-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".

I don't know a lot about topology, so I'm not sure how to understand/interpret this. In general, a Borel $\sigma $-algebra on a set $S$ is the $\sigma $-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".

Patricia Curry

Beginner2022-06-22Added 15 answers

For any family $\mathcal{F}$ of subsets of a set $X$ we can talk about the $\sigma $-algebra generated by $\mathcal{F}$ on $X$, which is the defined to be the intersection of all $\sigma $-algebras on $X$ that contain $\mathcal{F}$ as a subset. This is standard in measure theory. It's well-defined as the intersection of $\sigma $-algebras is a $\sigma $-algebra again and we always have at least one σ-algebra containing $\mathcal{F}$, namely the power set of $X$.

A topology on $X$ is just another such family $\mathcal{F}$ so the definition applies. The $\sigma $-algebra will thus contain all open sets, all closed sets (their complements), all ${G}_{\delta}$ sets and ${F}_{\sigma}$ sets etc. (all in that topology). It's also called the Borel $\sigma $-algebra of a topological space.

A topology on $X$ is just another such family $\mathcal{F}$ so the definition applies. The $\sigma $-algebra will thus contain all open sets, all closed sets (their complements), all ${G}_{\delta}$ sets and ${F}_{\sigma}$ sets etc. (all in that topology). It's also called the Borel $\sigma $-algebra of a topological space.