Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that $$\int_\mathbb{R}hd{\mu_f}=\int_{\mathbb{R}}hd\mu_g$$ for any Schwartz function $h$. I want to show that $\mu_f=\mu_g$.
Here there is a proof that the statement is true even if we restrict ourselves to test functions, but the proof looks quite complicated. Since Schwartz functions form a larger class, is there an easy proof for this statement?