Difference between revisions of "Borel sigma-algebra"

Latest revision as of 09:25, 6 August 2015

Note:

• Not to be confused with the Borel sigma-algebra generated by which, for a given topology $(X,\mathcal{O})$ is denoted $\mathcal{B}(X,\mathcal{J}):=\sigma(\mathcal{O})$ or just $\mathcal{B}(X)$ if the topology is implicit.
• (This page) The Borel $\sigma$-algebra refers to $\mathcal{B}(\mathbb{R})$, with it's usual topology (the topology induced by the absolute value metric, $\vert\cdot\vert$).
  • Because it is so common we simply denote it $\mathcal{B}$
  • Again, because it is so common, rather than saying a map is $\mathcal{A}/\mathcal{B}$-measurable, we may just say it is $\mathcal{A}$-measurable

Definition

The Borel $\sigma$-algebra is a^{[Note 1]} $\sigma$-algebra on $\mathbb{R}$^[1]. It is generated by the open sets of the metric space $(\mathbb{R},\vert\cdot\vert)$. We denote it as:

• $\mathcal{B}:=\sigma(\mathcal{O})$ where $\mathcal{O}$ denotes the open sets of $(\mathbb{R},\vert\cdot\vert)$^{[Note 2]}

This is actually a special case of the Borel $\sigma$-algebra generated by, rather than writing $\mathcal{B}(\mathbb{R})$ we simply write $\mathcal{B}$

The Borel $\sigma$-algebra can also be defined on $\mathbb{R}^n$, that is done as follows:

• $\mathcal{B}^n:=\mathcal{B}(\mathbb{R}^n)$^[1] with the usual topology on $\mathbb{R}^n$ (the metric given by the Euclidean norm will do)

Again, this is a special case of the Borel $\sigma$-algebra generated by a topology; this time it is the metric space $(\mathbb{R}^n,\vert\cdot\vert)$.

Generators

There are many generators of $\mathcal{B}^n$ (just use $n=1$ for $\mathcal{B}$ itself) - some are listed here. First here are some non-obvious definitions:

• $$[ [a,b))\subset\mathbb{R}^n$$ means $a$ and $b$ are $n$-tuples that denote the half-open-half-closed rectangles:
  • $$[ [a,b)):=[a_1,b_1)\times[a_2,b_2)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n$$ with the convention of:
    • $$[a_i,b_i)=\emptyset$$ if $b_i\le a_i$ and of course
    • $$[ [a,b))=\emptyset$$ if any of the $[a_i,b_i)=\emptyset$ - this is trivial to show.
  • The notation of $$((a,b)):=(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)$$ is similarly defined.
• $$\mathscr{J}_\text{rat}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{Q}^n\right\}$$
• $$\mathscr{J}_\text{rat}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{Q}^n\right\}$$
• $$\mathscr{J}^\circ:=\left\{((a,b))\vert\ a,b\in\mathbb{R}^n\right\}$$
• $$\mathscr{J}:=\left\{[ [a,b))\vert\ a,b\in\mathbb{R}^n\right\}$$

Also generated by:

TODO: Integrate these, also find proofs, they're just a remark in^[1]

Proof of claims

Notes

1. Jump up ↑ There are of course others, for example $\mathcal{P}(\mathbb{R})$ is always a $\sigma$-algebra but is much larger than the Borel one
2. Jump up ↑ Conventionally, $\mathcal{J}$ 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

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5} Measures, Integrals and Martingales - Rene L. Schilling