sigma-algebras as family of functions

Question

In textbooks the $\sigma$-algebra is always defined as a family of sets, and so for $\lambda$-system, $\pi$-system. But I just got a functional version of $\lambda$-system as:

The set $\mathcal L$ of bounded functions defined on a set $X$ is said to be a $\lambda$-system if:

1. $1\in \mathcal L$

2. $\mathcal L$ is a vector space

3. if $0\le f_n \uparrow f$, $f_n\in \mathcal L$ and $f$ is bounded then $f\in \mathcal L$

So I am wondering if $\sigma$-algebrad can also be defined in terms of functions.

Also I am wondering if there is a way to argue that the above definition is consistent with the version of $\lambda$-system defined in terms of sets.

Answer 1

There's a functional form of $\pi$-system as well: a class of bounded functions closed under point-wise multiplication. And with the vector space of bounded measurable functions playing the role of $\sigma$-algebra, even a functional monotone class theorem, immediately google-able.