Suppose that I have a function $f \colon \mathbb{R} \to \mathbb{R}$, such that for any random variable $X$ with mean $\mu$, and with invertible CDF, $\mathbb{E}[f(X)] \leq 0$.
Is it possible to give conditions on the function $f$ so that I can ensure that this inequality holds more generally for any random variable with mean $\mu$ and possibly noninvertible CDF?
One thought I had would possibly work for continuous $f$. Specifically, let $Y$ be some random variable with mean $\mu$, and such that its CDF is possibly not invertible. Perhaps one could construct random variables $Y_n$ each of which have invertible CDF, and have mean $\mathbb{E}[Y_n] = \mu$. Then, applying the inequality for $Y_n$, we would get $\mathbb{E}[f(Y_n)] \leq 0$, and therefore if say $f$ is continuous and $Y_n \to Y$ almost surely, then one would obtain $\mathbb{E}[Y] \leq 0$.
In this case, one would need to find a way to some how cook up $Y_n$ with invertible CDFs such that $Y_n \to Y$ almost surely, and $\mathbb{E}[Y_n] = \mu$.