Difference between revisions of "Borel sigma-algebra of the real line"

Latest revision as of 15:48, 27 February 2017

This page is a provisional page - see the notice at the bottom for more information

Definition

Let $(\mathbb{R},\mathcal{O})$^{[Note 1]} denote the real line considered as a topological space. Recall that the Borel $\sigma$-algebra is defined to be the $\sigma$-algebra generated by the open sets of the topology, recall that $\mathcal{J}$ is the collection of all open sets of the space. Thus:

• $\mathcal{B}(\mathbb{R}):\eq\sigma(\mathcal{O})$
  • where $\sigma(\mathcal{G})$ denotes the $\sigma$-algebra generated by $\mathcal{G}$, a collection of sets.

This is often written just as $\mathcal{B}$, provided this doesn't lead to ambiguities - this is inline with: $\mathcal{B}^n$, which we use for the Borel $\sigma$-algebra on $\mathbb{R}^n$

Other generators

Let $\mathbb{M}$ denote either the real numbers, $\mathbb{R}$, or the quotient numbers, $\mathbb{Q}$ (to save us writing the same thing for both $\mathbb{R}$ and $\mathbb{Q}$, then the following all generate^{[Note 2]} $\mathcal{B}(\mathbb{R})$:

1. $\{(-\infty,a)\ \vert\ a\in\mathbb{M}\}$^[1]
2. $\{(-\infty,b]\ \vert\ b\in\mathbb{M}\}$^[1]
3. $\{(c,+\infty)\ \vert\ c\in\mathbb{M}\}$^[1]
4. $\{[d,+\infty)\ \vert\ d\in\mathbb{M}\}$^[1]
5. $\{(a,b)\ \vert\ a,b\in\mathbb{M}\}$^[1]
6. $\{[c,d)\ \vert\ c,d\in\mathbb{M}\}$^[1]
7. $\{(p,q]\ \vert\ p,q\in\mathbb{M}\}$^{Suspected:[Note 3]}
8. $\{[u,v]\ \vert\ u,v\in\mathbb{M}\}$^{Suspected:[Note 4]}
9. $\mathcal{C}$^[1] - the closed sets of $\mathbb{R}$
10. $\mathcal{K}$^[1] - the compact sets of $\mathbb{R}$

Proofs

• 1, 2, 3 and 4: - the collection of all open and closed rays based at either rational or real points generate the Borel sigma-algebra on R
• 5: - the open rectangles with either rational or real points generate the same sigma-algebra as the Borel sigma-algebra on R^n
• 6: - the closed-open rectangles with either rational or real points generate the same sigma-algebra as the Borel sigma-algebra on R^n
• 7: - Warning:Suspected from proof on paper of $6$
• 8: - Warning:Suspected by^{[Note 4]}
• 9: - the sigma-algebra generated by the closed sets of R^n is the same as the Borel sigma-algebra of R^n
• 10: - the sigma-algebra generated by the compact sets of R^n is the same as the Borel sigma-algebra of R^n

See next

See also

Notes

1. Jump up ↑ Traditionally we use $\mathcal{J}$ for the topology part of a topological space, however later in the article we will introduce $\mathscr{J}$ in several forms, so we avoid $\mathcal{J}$ to avoid confusion.
2. Jump up ↑ This means that if $A$ is any of the families of sets from the list, then:
   • $\mathcal{B}(\mathbb{R})\eq\sigma(A)$.
3. Jump up ↑ I have proved form $6$ before, the order didn't matter there
4. ↑ ^{Jump up to: 4.0 4.1} Take: $$\bigcup_{n\in\mathbb{N} }[a+\frac{\epsilon}{n},b-\tfrac{\epsilon}{n}]$$ , with a little effort one can see this $\eq(a,b)$ - for carefully chosen $\epsilon$

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7} Measures, Integrals and Martingales - René L. Schilling