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

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Definition

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

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

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

Other generators

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

1. [ilmath]\{(-\infty,a)\ \vert\ a\in\mathbb{M}\} [/ilmath]^[1]
2. [ilmath]\{(-\infty,b]\ \vert\ b\in\mathbb{M}\} [/ilmath]^[1]
3. [ilmath]\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} [/ilmath]^[1]
4. [ilmath]\{[d,+\infty)\ \vert\ d\in\mathbb{M}\} [/ilmath]^[1]
5. [ilmath]\{(a,b)\ \vert\ a,b\in\mathbb{M}\} [/ilmath]^[1]
6. [ilmath]\{[c,d)\ \vert\ c,d\in\mathbb{M}\} [/ilmath]^[1]
7. [ilmath]\{(p,q]\ \vert\ p,q\in\mathbb{M}\} [/ilmath]^{Suspected:[Note 3] - almost certain}
8. Warning:May not be true: [ilmath]\{[u,v]\ \vert\ u,v\in\mathbb{M}\} [/ilmath]^{Suspected:[Note 4] - induced from pattern, unsure}
9. [ilmath]\mathcal{C} [/ilmath]^[1] - the closed sets of [ilmath]\mathbb{R} [/ilmath]
10. [ilmath]\mathcal{K} [/ilmath]^[1] - the compact sets of [ilmath]\mathbb{R} [/ilmath]

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 [ilmath]6[/ilmath]
• 8: - Warning:May not be true! note to self: the open balls are a basis (even at rational points with rational radiuses - countable basis) of [ilmath]\mathbb{R} [/ilmath], is there like a generator for closed sets?
• 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. ↑ Traditionally we use [ilmath]\mathcal{J} [/ilmath] for the topology part of a topological space, however later in the article we will introduce [ilmath]\mathscr{J} [/ilmath] in several forms, so we avoid [ilmath]\mathcal{J} [/ilmath] to avoid confusion.
2. ↑ This means that if [ilmath]A[/ilmath] is any of the families of sets from the list, then:
   • [ilmath]\mathcal{B}(\mathbb{R})\eq\sigma(A)[/ilmath].
3. ↑ I have proved form [ilmath]6[/ilmath] before, the order didn't matter there
4. ↑ I suspect this holds as the open balls basically are open intervals, sort of... anyway "it works" for the open balls, and the closed sets of [ilmath]\mathbb{R} [/ilmath] also generate [ilmath]\mathcal{B} [/ilmath] (see: form [ilmath]9[/ilmath]) so it might work

References

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