So I have been trying to apply Fourier transform to a particular setting/problem in probability theory. I am dealing with equations that do have logarithm of sigmoidal function or linear transformations of sigmoidal functions, where sigmoidal function is $\sigma(x):=(1+\exp(-x))$, and my functions are of the form $h_{\alpha}(x)\log(\alpha\sigma(x)+(1-\alpha))$ where these functions model conditional distribution of $p(y|x)$ where $y$ is a binary random variable.
As it stands since $h_{\alpha}(x)\not\in L_1$ it is not possible to conclude that Fourier transform of $h_{\alpha}(x)$ exists. On the other hand taking a density like Normal density would imply $h_{\alpha}(x)p(x)\in L_1$ which is just the expectation of conditional log-likelihood over $y=1$ examples, i.e. $\mathbb{E}_{p(x)}[\log(p(y=1|X)]$ which I presume(?), would enable me to apply Fourier transform on them.
Wikipedia refers to a claim that Fourier transform for a bounded function over a finite measure exist:
"By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead."
-- First question is that is this claim true and if so could you please refer me to a reference that proves this?
-- Second question is that in more general case where I have $h_{\alpha}(x)p(x)\in L_1$ can I use Fourier transform and then absorb $p(x)$ into $dx$ to convert Lebesgue measure to the measure of $p(x)$ and define my Fourier transform this way to make sure its existence?