structural:expansion, information:classification, linguistic:voice_change
Proof 2 (measure-theoretic form)

Let g {\displaystyle g} be a real-valued μ {\displaystyle \mu } -integrable function on a probability space Ω {\displaystyle \Omega }, and let φ {\displaystyle \varphi } be a convex function on the real numbers. Since φ {\displaystyle \varphi } is convex, at each real number x {\displaystyle x} we have a nonempty set of subderivatives, which may be thought of as lines touching the graph of φ {\displaystyle \varphi } at x {\displaystyle x}, but which are below the graph of φ {\displaystyle \varphi } at all points (support lines of the graph).

Now, if we define

because of the existence of subderivatives for convex functions, we may choose a {\displaystyle a} and b {\displaystyle b} such that

for all real x {\displaystyle x} and

But then we have that

for almost all ω ∈ Ω {\displaystyle \omega \in \Omega }. Since we have a probability measure, the integral is monotone with μ (Ω) = 1 {\displaystyle \mu (\Omega)=1} so that

as desired.