Generators for Extended Real Sigma-Algebra

Theorem

Let $\overline \BB$ be the extended real $\sigma$-algebra.

Then $\overline \BB$ is generated by each of the following collections of extended real intervals:

$\text {(1)}: \quad$

$\ds $

$\ds \set {\ \closedint a \to: a \in \R}$

$\text {(1')}: \quad$

$\ds $

$\ds \set {\ \closedint a \to: a \in \Q}$

$\text {(2)}: \quad$

$\ds $

$\ds \set {\ \hointl b \to: b \in \R}$

$\text {(2')}: \quad$

$\ds $

$\ds \set {\ \hointl b \to: b \in \Q}$

$\text {(3)}: \quad$

$\ds $

$\ds \set {\ \hointr \gets c: c \in \R}$

$\text {(3')}: \quad$

$\ds $

$\ds \set {\ \hointr \gets c: c \in \Q}$

$\text {(4)}: \quad$

$\ds $

$\ds \set {\ \closedint \gets d: d \in \R}$

$\text {(4')}: \quad$

$\ds $

$\ds \set {\ \closedint \gets d: d \in \Q}$

Proof

Let us first establish that $(1)$ up to $(4')$ all generate the same $\sigma$-algebra.

Denote $\GG_i$ for the collection at point $(i)$, and $\GG'_i$ for that at $(i')$, where $i = 1, 2, 3, 4$.

Furthermore, write $\Sigma_i$ for $\map \sigma {\GG_i}$ and $\Sigma'_i$ for $\map \sigma {\GG'_i}$.

Here $\sigma$ denotes generated $\sigma$-algebra.

By Generated Sigma-Algebra Preserves Subset, we have the following inclusions:

$\Sigma'_i \subseteq \Sigma_i$

for $i = 1, 2, 3, 4$.

Since we have the following observations about complements in $\overline \R$ (for arbitrary $a \in \R$):

$\relcomp {\overline \R} {\closedint a \to} = \hointr \gets a$

$\relcomp {\overline \R} {\closedint \gets a} = \hointl a \to$

we deduce that:

$\GG_3 \subseteq \Sigma_1, \GG'_3 \subseteq \Sigma'_1$

$\GG_2 \subseteq \Sigma_4, \GG'_2 \subseteq \Sigma'_4$

and by definition of generated $\sigma$-algebra:

$\Sigma_3 \subseteq \Sigma_1, \Sigma'_3 \subseteq \Sigma'_1$

$\Sigma_2 \subseteq \Sigma_4, \Sigma'_2 \subseteq \Sigma'_4$

By Complement of Complement, the converse inclusions:

$\Sigma_1 \subseteq \Sigma_3, \Sigma'_1 \subseteq \Sigma'_3$

$\Sigma_4 \subseteq \Sigma_2, \Sigma'_4 \subseteq \Sigma'_2$

are derived.

Subsequently, remark that, for all $a \in \R$:

$\closedint a \to = \ds \bigcap_{n \mathop \in \N_{>0}} \hointl {a - \frac 1 n} \to$

This article needs to be linked to other articles.
In particular: I know, but please find the justification yourself
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

and by Sigma-Algebra Closed under Countable Intersection, it follows that:

$\GG_1 \subseteq \Sigma_2, \GG'_1, \subseteq \Sigma'_2$

whence by definition of generated $\sigma$-algebra:

$\Sigma_1 \subseteq \Sigma_2, \Sigma'_1 \subseteq \Sigma'_2$

For the converse inclusion, remark that:

$\hointl a \to = \ds \bigcup_{n \mathop \in \N_{>0}} \closedint {a + \frac 1 n} \to$

This article needs to be linked to other articles.
In particular: again, justification
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

and thus immediately establish:

$\Sigma_2 \subseteq \Sigma_1, \Sigma'_2 \subseteq \Sigma'_1$

To summarize, the above arguments establish:

$\Sigma'_1 = \Sigma'_2 = \Sigma'_3 = \Sigma'_4 \subseteq \Sigma_4 = \Sigma_3 = \Sigma_2 = \Sigma_1$

Finally, for all $a \in \R$, we have:

$\hointl a \to = \ds \bigcup_{\substack {q \mathop \in \Q \\ q \mathop > a}} \hointl q \to$

This article, or a section of it, needs explaining.
In particular: via Between two Real Numbers exists Rational Number
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.