Jensen's Inequality (Measure Theory)/Convex Functions

Theorem
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \R$ be a $\mu$-integrable function such that $f \ge 0$ pointwise. Let $V: \left[{0 \,.\,.\, \infty}\right) \to \left[{0 \,.\,.\, \infty}\right)$ be a convex function.

Then for all positive measurable functions $g: X \to \R$, $g \in \mathcal{M}^+ \left({\Sigma}\right)$:


 * $V \left({\dfrac {\int g \cdot f \, \mathrm d \mu} {\int f \, \mathrm d \mu}}\right) \le \dfrac {\int \left({V \circ g}\right) \cdot f \, \mathrm d \mu} {\int f \, \mathrm d \mu}$

where $\circ$ denotes composition, and $\cdot$ denotes pointwise multiplication.