Borel $\sigma$-algebra of the real line

Definition

Let $(\mathbb{R},\mathcal{O})$ 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 $\mathcal{B}(\mathbb{R})$:

1. $\{(-\infty,a)\ \vert\ a\in\mathbb{M}\}$
2. $\{(-\infty,b]\ \vert\ b\in\mathbb{M}\}$
3. $\{(c,+\infty)\ \vert\ c\in\mathbb{M}\}$
4. $\{[d,+\infty)\ \vert\ d\in\mathbb{M}\}$
5. $\{(a,b)\ \vert\ a,b\in\mathbb{M}\}$
6. $\{[c,d)\ \vert\ c,d\in\mathbb{M}\}$
7. $\{(p,q]\ \vert\ p,q\in\mathbb{M}\}$^{Suspected: - almost certain}
8. Warning:May not be true: $\{[u,v]\ \vert\ u,v\in\mathbb{M}\}$^{Suspected: - induced from pattern, unsure}
9. $\mathcal{C}$ - the closed sets of $\mathbb{R}$
10. $\mathcal{K}$ - 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:May not be true! note to self: the open balls are a basis (even at rational points with rational radiuses - countable basis) of $\mathbb{R}$, 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

Grade: A*

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* Tidy up the proofs section, work on resolving 7 and especially 8, also

• maybe write $\mathcal{C}:\eq\{A\in\mathcal{P}(\mathbb{R})\ \vert\ A\text{ is closed }\}$ or something to give it a more clear definition for 9 and 10 Alec (talk) 22:15, 26 February 2017 (UTC)