Santino Bautista

2022-06-21

Reading about random sets, I've come across the phrase "$\mathcal{B}\left(\mathcal{F}\right)$ 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".

### Answer & Explanation

Patricia Curry

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.

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