How can I connect this "entropy functional" definition with information theoretic entropy?

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.

The entropy for a discrete random variable $$X$$ of law $$p$$ can be explicited as $$H(X) = - \sum_{i=1}^n p(x_i) \log (p(x_i)$$
On the other hand, if $$f$$ is a probability density function, one has $$E_\mu(f) = \int f d\mu = 1$$, hence $$\log (E_\mu(f)) = 0$$. One hence gets $$\operatorname{Ent}_\mu(f) = E_\mu(f \log (f))$$
edit (cf page 20 of the paper linked above) In fact, $$H(p) = -\operatorname{Ent}_\mu (p)$$, hence both definition coincide.