Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 8 / 3 / c / Solution

In the one-dimensional unit-mass setting, put $q=f/\gamma$. Using the convention $0\log 0=0$, the relative entropy isBecause $\int \gamma q=\int f=1=\int \gamma$, we can subtract $q-1$ inside the integral:This is relative entropy nonnegativity from the given scalar inequality. Its integrand vanishes only at $q=1$, so zero relative entropy characterizes $f=\gamma$ almost everywhere. The nonnegativity remains valid with value $+\infty$ when entropy is not finite.