In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by

1. $F(x)\equiv \mathrm{Pr}[X\leq x]$ for $x\in\mathbb R^n$
2. $\mathbb E[g(X)]$ for all continuous functions $g$
3. $\mathbb E[g(X)]$ for all bounded continuous functions $g$

Notably, characterization 3 weakens characterization 2 quite a bit in the sense that one only needs to know the expected value for a smaller set of functions $g(\cdot)$. My question is as follows: are there any results characterizing just how small the set of functions $g(\cdot)$ for which we need to know the expectations in order to characterize the distribution of $X$? Something like the Stone-Wierstrass approximation theorem seems to be related, but I am somewhat at a loss in terms of what to search for in connecting the dots.

In probability theory, there are a number of equivalent ways to characterize a distribution on $\mathbb R^n$. For example, the distribution of a random vector $X\in\mathbb R^n$ may be characterized by:

1. $F(x)\equiv \mathrm{Pr}[X\leq x]$ for $x\in\mathbb R^n$.
2. $\mathbb E[g(X)]$ for all continuous functions $g$.
3. $\mathbb E[g(X)]$ for all bounded continuous functions $g$.

Notably, characterization (3) weakens characterization (2) quite a bit in the sense that one only needs to know the expected value for a smaller set of functions $g(\cdot)$. My question is as follows: are there any results characterizing just how small the set of functions $g(\cdot)$ for which we need to know the expectations in order to characterize the distribution of $X$? Something like the Stone-Wierstrass approximation theorem seems to be related, but I am somewhat at a loss in terms of what to search for in connecting the dots.