This appeared in an exam I took.
The question asked us to give an example of a convex function $g: \mathbb{R} \longmapsto \mathbb{R}$ and a measure $\mu$ on $\left(\mathbb{R}, \mathscr{B}(\mathbb{R})\right)$ such that $g\left(\int x \, d\mu(x)\right) > \int g(x)\, d\mu(x)$.
I am assuming that such an example can be constructed by violating the finiteness of the measure, but I have no idea how I would construct such an example. I guess the Lebesgue measure can be used, but am having trouble finding a function that is convex and gives this result.