Jensen's Inequality (Measure Theory)/Concave Functions

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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 $\Lambda: \left[{0 \,.\,.\, \infty}\right) \to \left[{0 \,.\,.\, \infty}\right)$ be a concave function.


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

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

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


Proof