Let $\mu$ be a positive measure on a measure space $\Omega$. How can I show that for all non-negative measurable $f$ $$\exp\left(\int_\Omega f \; d\mu \right) \leq \int_\Omega \exp(f) \; d\mu$$ the above inequality is valid if $\mu(\Omega) = 1$ and not valid if $\mu(\Omega) \neq 1.$
This is what I've been able to come up with.
Since $\exp$ is convex, if $\mu(\Omega) =1$, then the inequality is just Jensen's inequality. Do I have to say more? How do I show that the inequality is not valid if $\mu(\Omega) \neq 1$.