Question: Suppose that $f$ and $g$ are two probability density functions, show that $\int g\log (g/f)$ is always non-negative and equals to $0$ $\it only\ if$ $\ g=f$ almost everywhere.
I have two questions here. First, what does it mean by "only if", please? Should I prove from the former to the latter or from the latter to the former, please? Second, what does it mean by "almost everywhere" and what is the implication for this particular question, please? Thank you!
It means for every two functions $f$ and $g$ with $f\neq g$, $\int g\log (g/f)$ is non-zero.
Here almost everywhere means for all sub-sets of the sample space which can be assigned a positive measure.