By the law of the unconscious statistician, the expected value of the function $t(x)=e^{-i\omega x}$ when x is distributed with probability density function $f(x)$ is$$E(e^{-i\omega x})=\int^{+\infty}_{-\infty}{e^{-i\omega x}f(x)dx}=\mathcal{F(f(x))(\omega)}$$ In fact, all density functions are absolutely integrable, so their Fourier transform exists.
I have not been able to find more information about this (there are too many search results about the expected value of the Fourier series). Is there in general a way to interpret Fourier transforms as the collection of expected values of oscillating random variables? Is this used anywhere? I imagine this could be used to approximate an unknown distribution by doing the inverse transform of a bunch of averages of $e^{-i\omega x_i}$, with $x_i$ coming from that distribution.
I'd be thankful if anyone could share any insight into this, or sources to read more about it.