One very basic question regarding the topology of weak convergence.
We know that given the following:
$X$ metrizable topological space,
$\mathcal{B} (X)$ Borel $\sigma$-algebra,
$\Delta (X)$ set of all probability measures over $\mathcal{B} (X)$,
$C_b (X)$ set of all continuous bounded functionals on $X$,
the $w^*$ topology on $\Delta (X)$, called the topology of weak convergence, is such that, for all $f \in C_b (X)$, $\mu \mapsto \int f \,d \mu$ is $w^*$-continuous.
The question is, what do the $f \in C_b (X)$ stand for in terms of probabilistic intuition?
Should I see them as random variables or what?
Thank you for your time.
Well, if we take $C_b(X)$ as a subset of the random variables, then the map $\mu \mapsto \int f d\mu$ is of course taking the expected value. If we were to try to evaluate that on on all random variables, there would be, of course, random variables for which this is undefined. Indeed, for any measure there would be random variables where the evaluation was undefined.
So, regarding them as all the random variables is clearly wrong. But they are an algebra of random variables (closed under multiplication) where the expected values necessarily exist. They aren't the largest such algebra, but they are a large enough collection to make convergence meaningful.