 Relative entropy nonnegativity

For probability densities $f,g$ with $g>0$, set $q=f/g$. Equal masses give $H(f\mid g)=\int g(q\log q-q+1)\ge 0$. The scalar integrand is minimized only at $q=1$, so equality means $f=g$ almost everywhere. Use $0\log 0=0$; infinite entropy is allowed.