We're talking about th eMonotone Class Theorem in my probability course and I've noticed some similarities with the Stone-Weierstrass Theorem. I'm told there's a number of different versions of the Monotone Class Theorem, so let me state the one we're using.
Suppose $\cal{H}$ is a class of bounded functions $\Omega \rightarrow \mathbb{R} $ with the following properties:
$\bullet \cal{H} \text{ is a real vector space} $
$\bullet \cal{H} \text{contains the constant functions}$
$\bullet \text{ if } (f_n)_n \text{ is an increasing sequence of functions in } \cal{ H } \text{ and } f_n \rightarrow f \text{ where } f \text{ is also bounded, then } f \in \cal{H}$
Assume $\cal{C} \subset \cal{H}$ is closed under pointwise multiplication. Then $\cal{H}$ contains all $\sigma({\cal{C}})$- measurable functions.
Aparat from the obvious similarities in the statement, the proof we're given also resembles quite a lot the proof of Stone Weierstrass - we're essentially proving that it suffices to consider the case when $\cal{C}$ is a closed unital algebra of functions - using the same kind of approximating $|f|$ by polynomials in $f$ argument- and then showing for such $\cal{C}$ all $\sigma(\cal{C})$-measurable functions are already in $\cal{C}$. I'm thus wondering if there is maybe some general statement (possibly even categorical) that encapsulates both these results. Has anyone encountered something like this before?