Let $X$ be a random variable, is it true that I can find $f_n \in C_c$ such that $E[f_n(X)] \uparrow E[f(X)]$, where $f$ is absolutely continuous and $E[f(X)] < \infty$? I'm a little concerned since I'm dealing with Riemann-Stieljes integral. Any pointer is welcomed. Thank you.
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Yes. The Dominated Convergence theorem works for arbitrary measure space. Intuitively you can take an indicitator function on $[-n,n]$ and take $n$ to infinity. To make it an absolutely continuous on all of the real line, convolve with a smooth function to get a mollifier. To ensure a compactly supported mollifier, you need to ensure the smooth function you are convolving with is also compactly supported. There is an example in the link of such a function.