Jensen's Inequality (Measure Theory)/Concave Functions

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f: X \to \R$ be a $\mu$-integrable function such that $f \ge 0$ pointwise. Let $\Lambda: \hointr 0 \infty \to \hointr 0 \infty$ be a concave function.

Then for all positive measurable functions $g: X \to \R$, $g \in \map {\MM^+} \Sigma$:


 * $\dfrac {\int \paren {\Lambda \circ g} \cdot f \rd \mu} {\int f \rd \mu} \le \map \Lambda {\dfrac {\int g \cdot f \rd \mu} {\int f \rd \mu} }$

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