Suppose $\mu$ is a probability measure on $S$ and $f:S\to {\mathbf{R}}_0^{+}$ is such that $\int fd\mu =1$. Then $$\int |f-1|d\mu \le \sqrt{-2\mathrm{E}\mathrm{n}\mathrm{t}(f)}.$$ This is called Pinsker's inequality. Hint: Use the elementary inequality: $3(x-1{)}^2\le (4+2x)(x\log x-x+1)$, which holds for all $x\ge 0$. Thus if $\mathrm{E}\mathrm{n}\mathrm{t}(P^{n}f)$ converges to $0$, then $P^{n}f$ converges in $L_1(\mu )$ to the constant function $1$.