Borel sigma-algebra generated by
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Definition
The Borel [ilmath]\sigma[/ilmath]-algebra is the [ilmath]\sigma[/ilmath]-algebra generated by the open sets of a topological space, that is[1]: (where [ilmath](X,\mathcal{O})[/ilmath][Note 1] is any topology)
- [ilmath]\mathcal{B}(X,\mathcal{O}):=\sigma(\mathcal{O})[/ilmath] - if the topology on [ilmath]X[/ilmath] is obvious, we may simply write: [ilmath]\mathcal{B}(X)[/ilmath][1]
Generators
For a topological space [ilmath](X,\mathcal{O})[/ilmath] the following can be shown:
| Claim | Proof route | Comment |
|---|---|---|
| [ilmath]\mathcal{B}(X):=\sigma(\mathcal{O})[/ilmath] | Trivial (by definition) | |
| [ilmath]\mathcal{B}(X)=\sigma(\mathcal{C})[/ilmath] - the closed sets | [ilmath]\mathcal{B}(X):=\sigma(\mathcal{O})=\sigma(\mathcal{C})[/ilmath] - see claim 1 below |
Proof of claims
Claim 1: [ilmath]\sigma(\mathcal{O})=\sigma(\mathcal{C})[/ilmath]
TODO: Be bothered, just use complements
See also
- Borel [ilmath]\sigma[/ilmath]-algebra - a special case, where [ilmath]\mathcal{B}:=\mathcal{B}(\mathbb{R},\vert\cdot\vert)[/ilmath] and [ilmath]\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n,\vert\cdot\vert)[/ilmath]
Notes
- ↑ Note the letter [ilmath]\mathcal{O} [/ilmath] for the open sets of the topology, conventionally [ilmath]\mathcal{J} [/ilmath] is used, however in measure theory this notation is often used to denote the set of half-open-half-closed rectangles in [ilmath]\mathbb{R}^n[/ilmath] - a totally separate thing
References
