Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $$ whenever the left hand side is defined.

If it helps, I know that the $\log$ function is concave.

Can I argue like this: Since $\log$ concave, $-\log$ is convex; So by Jensen's inequality, we have $$ -\log~\int g(u)~\text{d}u ~\leq~ \int-\log ~(g(u))~\text{d}u?$$

  • $\begingroup$ Presumably, you are considering a version of $\log$ that takes values in the extended real line? $\endgroup$
    – cardinal
    Nov 10 '11 at 16:52
  • $\begingroup$ Yes, Jensen's inequality can be used, in the opposite direction, with concave functions, just as you've shown in your Edit. $\endgroup$
    – robjohn
    Nov 10 '11 at 17:46
  • $\begingroup$ A much nicer problem is to show that $\lim_{p \to 0} \|f\|_{L^p} = \exp \left ( \int_X \log |f(x)| \, \textrm{d}\mu(x) \right )$. $\endgroup$
    – JT_NL
    Nov 10 '11 at 18:06

Looks like just an instance of Jensen's inequality to me (unless maybe there's a question about whether it's the left side or the right side that is assumed to be defined?).


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