Let $p,q$ be two distributions on $\mathbb{R}^d$. Let $f:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$. Under what conditions does $\mathbb{E}_{x\sim p}f(x,z)=\mathbb{E}_{x\sim q}f(x,z)\ \forall z\in\mathbb{R}^d$ indicate $p=q$ a.e.? For example, can we prove $p=q$ a.e. when $f(x,z)=\tanh(x^{\top}z)$?

(Note: this problem may be related to discriminatory functions such as sigmoidal functions, but two distributions are involved in this case.)

nosince $\tanh xz$ is odd, so the integral will be zero for all $z$ as soon as $p-q$ is even (so in particular, for any two even $p,q$). $\endgroup$ – Christian Remling Oct 10 at 14:45