This might seem very obvious, but I am thrown for a loop, and a search of StackExchange did not help. In this paper, we have the definition
The entropy $\text{Ent}_\mu(f)$ of a $\mu$-integrable function $f$ is defined to be $\text{Ent}_\mu(f):= E_\mu(f \log f) - E_\mu(f) \log[E_\mu f ] $.
This doesn't seem to match the information theoretic definition of entropy which I am familiar with, where with probability measure $p$, we have
$H_p(X) = E_p[ -\log[p(X)] ]$.
This doesn't line up if we take some trivial case where our measure $\mu$ is just $p$, and $f$ is just the random variable $X$. What am I missing here? Are these referring to the same concept in two different contexts? I'm very weak at understanding the measure theory notation, and I'm struggling to understand this.