I came across this question from Rudin's Real and Complex Analysis, 3rd Edition (p.75 # 25)
"Suppose that $\mu$ is a positive measure on $X$ and $f:X\rightarrow (0,\infty)$ satisfies $\int_X f d\mu = 1$. Prove for every $E \subset X$ with $0 < \mu(E) < \infty$ that $\int_E(\log{f})d\mu \leq \mu(E)\log{\frac{1}{\mu(E)}} $
Also, when $0
, we have $\int_E{f^p}d\mu \leq \mu(E)^{1-p}$."
My first thought was that since log is a concave function, we can use Jensen's inequality (in the opposite direction), but that is not giving me what I want. Any suggestions?
Further Addendum : Jensen's inequality only works on a set of measure 1 (or by redefining an interval to get a measure 1 set) so this is clearly not the correct approach.