I want to prove the expectation of the function $f(X)$ exists, where $X$ is a Gaussian variable. Given that $f(X)$ is bounded between $[-c, c]$, with $c$ being some positive constant, does the expectation of $E[f(x)]$ always exist?
I think it exists because $$E[f(x)]=\int_{-\infty}^\infty f(x)\frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2}/(2\sigma^2),$$ and since $-c\leq f(x)\leq c$ for any $x$, $$\int_{-\infty}^\infty -c\frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2}/(2\sigma^2)\leq E[f(x)]\leq\int_{-\infty}^\infty c\frac{1}{\sqrt{2\pi}\sigma}e^{-(x-\mu)^2}/(2\sigma^2).$$ Those two integrations are bounded so $E[f(x)]$ is bounded. Am I correct?
Looks good to me. By this same reasoning you can say $-c \leq E\{ f(x) \} \leq c$, which is even stronger.