Jensen inequality in measure theory : why doesn't the convex function need to be nonnegative?

Question

This section of the Wikipedia article on Jensen's inequality states that if $g$ is an integrable function on a measure space with mass $1$ and $\varphi$ is a convex function, then $$\varphi \left( \int g \right) \leq \int \varphi \circ g $$ What troubles me is that I can see no mention of a hypothesis ensuring that $\varphi \circ g$ actually has an integral (like that it is integrable or else $\varphi$ is nonnegative). I assumed it was simply missing, but then I noticed that not only does the same article in French also omit these hypotheses but furthermore, it even bothers to mention that the integral on the right may be infinite, suggesting in my opinion that $\varphi$ should be assumed to be nonnegative.

Answer 1

$\varphi \circ g$ always has integrable negative part so that the integral on the right hand side is well-defined via $$\int \varphi \circ g d \mu = \int (\varphi \circ g)^+ d\mu - \int(\varphi \circ g)^- d\mu$$

To prove this, note that convex functions possess subderivatives so that there are numbers $a,b$ such that $$ax + b \leq \varphi(x)$$

In particular, $(\varphi \circ g) \geq a g + b$ so that $$0 \leq (\varphi \circ g)^{-} \leq |a g + b| \in L^1$$