Borel [ilmath]\sigma[/ilmath]-algebra of the real line
- This page is a provisional page - see the notice at the bottom for more information
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]:
- [ilmath]\{(-\infty,a)\ \vert\ a\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{(-\infty,b]\ \vert\ b\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{(c,+\infty)\ \vert\ c\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{[d,+\infty)\ \vert\ d\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{(a,b)\ \vert\ a,b\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{[c,d)\ \vert\ c,d\in\mathbb{M}\} [/ilmath]^{[1]}
- [ilmath]\{(p,q]\ \vert\ p,q\in\mathbb{M}\} [/ilmath]^{Suspected:}^{[Note 3]}
- [ilmath]\{[u,v]\ \vert\ u,v\in\mathbb{M}\} [/ilmath]^{Suspected:}^{[Note 4]}
- [ilmath]\mathcal{C} [/ilmath]^{[1]} - the closed sets of [ilmath]\mathbb{R} [/ilmath]
- [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: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
The message provided is:
See next
See also
Notes
- ↑ 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.
- ↑ 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].
- ↑ I have proved form [ilmath]6[/ilmath] before, the order didn't matter there
- ↑ ^{4.0} ^{4.1} Take: [math]\bigcup_{n\in\mathbb{N} }[a+\frac{\epsilon}{n},b-\tfrac{\epsilon}{n}][/math], with a little effort one can see this [ilmath]\eq(a,b)[/ilmath] - for carefully chosen [ilmath]\epsilon[/ilmath]
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} ^{1.6} ^{1.7} Measures, Integrals and Martingales - René L. Schilling
- Pages requiring work
- Provisional pages of grade: A*
- Provisional pages
- Definitions
- Measure Theory Definitions
- Measure Theory
- Analysis Definitions
- Analysis
- Functional Analysis Definitions
- Functional Analysis
- Theorems
- Theorems, lemmas and corollaries
- Measure Theory Theorems
- Measure Theory Theorems, lemmas and corollaries
- Analysis Theorems
- Analysis Theorems, lemmas and corollaries
- Functional Analysis Theorems
- Functional Analysis Theorems, lemmas and corollaries