Jensen's Inequality (Measure Theory)/Convex 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 $V: \hointr 0 \infty \to \hointr 0 \infty$ be a convex function.

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


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

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

Proof
Let $\d \map \nu x := \dfrac {\map f x} {\int \map f s \rd \map \mu s} \rd \map \mu x$ be a probability measure.

Let $x_0 := \displaystyle \int \map g s \rd \map \nu s$.

Then by convexity there exists constants $a, b$ such that $\forall x \in \R_{\ge 0}$:


 * $\map V {x_0} = a x_0 + b$


 * $\map V x \ge a x + b$

In other words, there is a tangent line at $\tuple {x_0, V_0}$ that falls below the graph of $V$.

Therefore: