Generators for Extended Real Sigma-Algebra

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Theorem

Let $\overline{\mathcal B}$ be the extended real $\sigma$-algebra.


Then $\overline{\mathcal B}$ is generated by each of the following collections of extended real intervals:

\((1):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{a \,.\,.\, +\infty}\right]: a \in \R}\right\}\)
\((1'):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{a \,.\,.\, +\infty}\right]: a \in \Q}\right\}\)
\((2):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left({b \,.\,.\, +\infty}\right]: b \in \R}\right\}\)
\((2'):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left({b \,.\,.\, +\infty}\right]: b \in \Q}\right\}\)
\((3):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{-\infty \,.\,.\, c}\right): c \in \R}\right\}\)
\((3'):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{-\infty \,.\,.\, c}\right): c \in \Q}\right\}\)
\((4):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{-\infty \,.\,.\, d}\right]: d \in \R}\right\}\)
\((4'):\quad\) \(\displaystyle \) \(\) \(\displaystyle \left\{ {\ \left[{-\infty \,.\,.\, d}\right]: d \in \Q}\right\}\)


Proof

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

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

Furthermore, write $\Sigma_i$ for $\sigma \left({\mathcal G_i}\right)$ and $\Sigma'_i$ for $\sigma \left({\mathcal G'_i}\right)$.

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 $a \in \R$):

$\complement_{\overline \R} \left({\left[{a \,.\,.\, +\infty}\right]}\right) = \left[{-\infty \,.\,.\, a}\right)$
$\complement_{\overline \R} \left({\left[{-\infty \,.\,.\, a}\right]}\right) = \left({a \,.\,.\, +\infty}\right]$

we deduce that:

$\mathcal G_3 \subseteq \Sigma_1, \mathcal G'_3 \subseteq \Sigma'_1$
$\mathcal G_2 \subseteq \Sigma_4, \mathcal G'_2 \subseteq \Sigma'_4$

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 $a \in \R$:

$\left[{a \,.\,.\, +\infty}\right] = \displaystyle \bigcap_{n \mathop \in \N} \left({a - \frac 1 n \,.\,.\, +\infty}\right]$

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

$\mathcal G_1 \subseteq \Sigma_2, \mathcal G'_1, \subseteq \Sigma'_2$

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

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

For the converse inclusion, remark that:

$\left({a \,.\,.\, +\infty}\right] = \displaystyle \bigcup_{n \mathop \in \N} \left[{a + \frac 1 n \,.\,.\, +\infty}\right]$

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:

$\left({a \,.\,.\, +\infty}\right] = \displaystyle \bigcup_{\substack{q \mathop \in \Q \\ q \mathop > a}} \left({q \,.\,.\, +\infty}\right]$


whence $\Sigma_2 \subseteq \Sigma'_2$, and all eight $\sigma$-algebras are equal; denote them by $\Sigma$ from now on.


It remains to establish that in fact they equal $\overline{\mathcal B}$.

Since all elements of $\mathcal G_2$ are open sets in the extended real number space, it follows that:

$\Sigma_2 \subseteq \overline{\mathcal B}$


By Intervals with Extended Rational Endpoints form Countable Basis for Extended Real Number Space, every open set in the extended real number space is of the form:

$\displaystyle \bigcup \mathcal Q$

where $\mathcal Q$ is a collection of extended real intervals with extended rational endpoints.

By Set of Intervals with Extended Rational Endpoints is Countable, $\mathcal Q$ is a subset of a countable set.

By Subset of Countably Infinite Set is Countable, $\mathcal Q$ is countable as well.


It follows by $\sigma$-algebra axiom $(3)$ that it suffices to show that:

$\left({a \,.\,.\, b}\right) \in \Sigma$
$\left({a \,.\,.\, +\infty}\right] \in \Sigma$
$\left[{-\infty \,.\,.\, b}\right) \in \Sigma$
$\left[{-\infty \,.\,.\, +\infty}\right) \in \Sigma$

for all rational numbers $a, b \in \Q$.

The middle two are in $\Sigma'_2$ and $\Sigma'_4$, respectively, hence in $\Sigma$.

By Sigma-Algebra Closed under Intersection:

$\left({a \,.\,.\, b}\right) = \left({a \,.\,.\, +\infty}\right] \cap \left[{-\infty \,.\,.\, b}\right) \in \Sigma$

and by Sigma-Algebra Closed under Union:

$\left({-\infty \,.\,.\, +\infty}\right) = \left[{a \,.\,.\, +\infty}\right] \cup \left[{-\infty \,.\,.\, a}\right] \in \Sigma$


Hence $\Sigma = \overline{\mathcal B}$, as desired.

$\blacksquare$


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