Borel sigma-algebra
Note:
- Not to be confused with the Borel sigma-algebra generated by which, for a given topology [ilmath](X,\mathcal{O})[/ilmath] is denoted [ilmath]\mathcal{B}(X,\mathcal{J}):=\sigma(\mathcal{O})[/ilmath] or just [ilmath]\mathcal{B}(X)[/ilmath] if the topology is implicit.
- (This page) The Borel [ilmath]\sigma[/ilmath]-algebra refers to [ilmath]\mathcal{B}(\mathbb{R})[/ilmath], with it's usual topology (the topology induced by the absolute value metric, [ilmath]\vert\cdot\vert[/ilmath]).
- Because it is so common we simply denote it [ilmath]\mathcal{B} [/ilmath]
- Again, because it is so common, rather than saying a map is [ilmath]\mathcal{A}/\mathcal{B} [/ilmath]-measurable, we may just say it is [ilmath]\mathcal{A} [/ilmath]-measurable
Definition
The Borel [ilmath]\sigma[/ilmath]-algebra is a[Note 1] [ilmath]\sigma[/ilmath]-algebra on [ilmath]\mathbb{R} [/ilmath][1]. It is generated by the open sets of the metric space [ilmath](\mathbb{R},\vert\cdot\vert)[/ilmath]. We denote it as:
- [ilmath]\mathcal{B}:=\sigma(\mathcal{O})[/ilmath] where [ilmath]\mathcal{O} [/ilmath] denotes the open sets of [ilmath](\mathbb{R},\vert\cdot\vert)[/ilmath][Note 2]
This is actually a special case of the Borel [ilmath]\sigma[/ilmath]-algebra generated by, rather than writing [ilmath]\mathcal{B}(\mathbb{R})[/ilmath] we simply write [ilmath]\mathcal{B} [/ilmath]
The Borel [ilmath]\sigma[/ilmath]-algebra can also be defined on [ilmath]\mathbb{R}^n[/ilmath], that is done as follows:
- [ilmath]\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n)[/ilmath][1] with the usual topology on [ilmath]\mathbb{R}^n[/ilmath] (the metric given by the Euclidean norm will do)
Again, this is a special case of the Borel [ilmath]\sigma[/ilmath]-algebra generated by a topology; this time it is the metric space [ilmath](\mathbb{R}^n,\vert\cdot\vert)[/ilmath].
Generators
There are many generators of [ilmath]\mathcal{B}^n[/ilmath] (just use [ilmath]n=1[/ilmath] for [ilmath]\mathcal{B} [/ilmath] itself) - some are listed here. First here are some non-obvious definitions:
- [math][ [a,b))\subset\mathbb{R}^n[/math] means [ilmath]a[/ilmath] and [ilmath]b[/ilmath] are [ilmath]n[/ilmath]-tuples that denote the half-open-half-closed rectangles:
- [math][ [a,b)):=[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n[/math] with the convention of:
- [math][a_i,b_i)=\emptyset[/math] if [ilmath]b_i\le a_i[/ilmath] and of course
- [math][ [a,b))=\emptyset[/math] if any of the [ilmath][a_i,b_i)=\emptyset[/ilmath] - this is trivial to show.
- The notation of [math]((a,b)):=(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)[/math] is similarly defined.
- [math][ [a,b)):=[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n[/math] with the convention of:
- [math]\mathscr{J}_\text{rat}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{Q}^n\right\}[/math]
- [math]\mathscr{J}_\text{rat}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{Q}^n\right\}[/math]
- [math]\mathscr{J}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{R}^n\right\}[/math]
- [math]\mathscr{J}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{R}^n\right\}[/math]
| Claim | Proof route | Comment |
|---|---|---|
| [ilmath]\mathcal{B}^n:=\sigma(\mathcal{O})[/ilmath] - open[1] | Trivial (by definition) | |
| [ilmath]\mathcal{B}^n=\sigma(\mathcal{C})[/ilmath] - closed[1] | Showing [ilmath]\sigma(\mathcal{C})\subseteq\sigma(\mathcal{O})[/ilmath] and [ilmath]\sigma(\mathcal{O})\subseteq\sigma(\mathcal{C})[/ilmath] - see Claim 1 | This is true for any Borel [ilmath]\sigma[/ilmath]-algebra generated by a topology |
| [ilmath]\mathcal{B}^n=\sigma(\mathcal{K})[/ilmath] - compact[1] |
TODO: There's quite a few steps and theorems required (eg: compact set in Hausdorff space is closed) |
Link with generated borel sigma algebra - which requires a Hausdorff metric space I believe |
| [ilmath]\mathcal{B}^n=\sigma(\mathcal{O})=\sigma(\mathscr{J}_\text{rat}^\circ)=\sigma(\mathscr{J}^\circ)[/ilmath] | Claim 2 | |
|
TODO: Check this and the method from the book - page 67 of my notes |
Also generated by:
| [ilmath]\{(-\infty,a)\vert\ a\in\mathbb{Q}\}[/ilmath] | [ilmath]\{(-\infty,a)\vert\ a\in\mathbb{R}\}[/ilmath] |
| [ilmath]\{(-\infty,b]\vert\ b\in\mathbb{Q}\}[/ilmath] | [ilmath]\{(-\infty,b]\vert\ b\in\mathbb{R}\}[/ilmath] |
| [ilmath]\{(c,+\infty)\vert\ c\in\mathbb{Q}\}[/ilmath] | [ilmath]\{(c,+\infty)\vert\ c\in\mathbb{R}\}[/ilmath] |
| [ilmath]\{[d,+\infty)\vert\ c\in\mathbb{Q}\}[/ilmath] | [ilmath]\{[d,+\infty)\vert\ c\in\mathbb{R}\}[/ilmath] |
TODO: Integrate these, also find proofs, they're just a remark in[1]
Proof of claims
Claim 1: [ilmath]\sigma(\mathcal{O})=\sigma(\mathcal{C})[/ilmath]
TODO: Be bothered, just use complements
Claim 2: Todo - even write this
TODO: This
Notes
- ↑ There are of course others, for example [ilmath]\mathcal{P}(\mathbb{R})[/ilmath] is always a [ilmath]\sigma[/ilmath]-algebra but is much larger than the Borel one
- ↑ Conventionally, [ilmath]\mathcal{J} [/ilmath] denotes the open sets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it is too common to ignore
