# How to think of a sub-sigma algebra as a collection of random variables?

Where $$\mathcal{H}$$ is a $$\sigma$$-algebra on $$\Omega$$, section 9.1 here discusses thinking of a sub-$$\sigma$$-algebra $$\mathcal{G}$$ of $$\mathcal{H}$$ "as the collection of all numerical random variables that are $$\mathcal{G}$$-measurable".

This answer gives some intuition for $$\sigma$$-algebras generated by random variables ("restricting the world's probabilism"), which aligns with the lecture notes (the first link). Even so, I don't see where any functions (random variables) are in a set of sets ($$\mathcal{G}$$).

Can I have some pointers to further reading?

• A random variable determines a sub sigma algebra (its events are the inverse images of the measurable sets of real numbers). Conversely, a sub sigma algebra determines the space of all measurable functions: the random variables. The correspondence is one-to-one.
– whuber
Commented Jan 28, 2023 at 14:58

$$\newcommand{\classG}{\mathcal{G}}$$

To facilitate discussion, let's first settle the probability space to be $$(\Omega, \mathcal{H}, P)$$, and let $$\mathcal{G}$$ be a sub-$$\sigma$$-algebra of $$\mathcal{H}$$. The complete sentence that the author wrote is:

Cinlar advocates for thinking of the sub-$$\sigma$$-algebra $$\mathcal{G}$$ both as a collection of events (i.e., measurable subsets of $$\Omega$$), and as the collection of all numerical random variables that are $$\classG$$-measurable.

The first part of the quote is obvious: by definition $$\classG$$ (in fact any subclass $$\mathcal{A}$$ of $$\mathcal{H}$$) is a collection of events.

Mechanically speaking, the second part of the quote does not make sense, because $$\classG$$ is a class of sets, instead of random variables (i.e., functions defined on $$\Omega$$). However, such "thinking" does make sense in view of the following isomorphism $$\tau$$ between $$\classG$$ and $$\mathscr{I}$$, which is defined as the class of indicator functions of events in $$\classG$$: for every $$G \in \classG$$, define $$\tau(G) = I_G(\omega), \omega \in \Omega$$. Now $$\mathscr{I} = \{I_G: G \in \classG\}$$ is a collection of numerical random variables that are $$\classG$$-measurable -- intuitively, for each $$\omega \in \Omega$$, knowing the value of $$I_G(\omega)$$ is equivalent to knowing if the event $$G$$ occurs, that is, if $$\omega \in G$$. In addition, $$\tau$$ is obviously a bijective between $$\classG$$ and $$\mathscr{I}$$. This justifies the second part of the quote.

For more enlightening discussion on sub-$$\sigma$$-algebra and its heuristic relation to partial information, I recommend you reading the Subfields subsection in Section 4 of Probability and Measure by Patrick Billingsley.

• To clarify, $\mathscr{I}$ doesn't have all $\mathcal{G}$-measurable numerical random variables, right? Commented Jan 29, 2023 at 12:32
• Right, $\mathscr{I}$ cannot contain all $\mathcal{G}$-measurable functions. But there is a theorem that any $\mathcal{G}$-measurable function is a limit of simple functions (which are finite linear combinations of members in $\mathscr{I}$). So to strictly fit the qualifier all, you may enlarge the definition of $\mathscr{I}$ to the monotone class generated by all simple functions. However, this loses the transparent 1-to-1 correspondence between $\mathcal{G}$ and $\mathscr{I}$. To me, the statement made by the author is largely heuristic, which is probably not mathmatically provable. Commented Jan 29, 2023 at 13:26
• @johnsmith The statement is more relevant if $\mathcal{G} = \sigma(X)$ (where $X$ is a given random variable), for which $\mathscr{I}$ can be modified to $\mathscr{I} = \{f(X): f \text{ is a Borel function from } \mathbb{R} \to \mathbb{R}\}$. In this case $\mathscr{I}$ indeed contains all $\mathcal{G}$-measurable functions since any $\sigma(X)$-measurable function must be a function of $X$. Commented Jan 31, 2023 at 1:57