I am looking for a reference for the following claim:
Let $X$ be a probability space, and let $g:X \to \mathbb [0,\infty) $ be in $L^1(X)$. Let $\phi:\mathbb [0,\infty) \to [0,\infty)$ be convex and strictly convex on $[a,\infty)$, for some $a \in (0,\infty)$, and suppose that $\int_X \phi\circ g=\phi(\int_X g)$.
Then, $\int_X g \in [a,\infty)$ implies that $g$ is constant a.e. , and $\int_X g \in (0,a]$ implies that $g \le a$ a.e..
I tried looking at "Real analysis" by Royden, and "Real and complex analysis" by Rudin but couldn't find such things. (not even in the exercises).
I guess I need to look at some book on convex analysis, but I am not familiar with the literature.
Is there any book (or paper) which contains such variants of Jensen?