Let $p$ and $q$ be two probability density functions, consider the identity:
$$ \int \left[\log(p(x))\right]p(x)dx = \int \left[\log(p(x))\right]q(x)dx. \qquad(*) $$
Assuming both integrals exist, under what conditions can we conclude that identity $(*)$ implies that $p$ and $q$ are identical?
Some thoughts: if $p$ is the uniform i.e. $p(x)=c$ over some bounded set, then the $(*)$ is true for all $p$ and $q$. But I think that is the only case.