Jensen's Inequality (Measure Theory)
This proof is about Jensen's Inequality in the context of Measure Theory. For other uses, see Jensen's Inequality.
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.
Convex Functions
Let $V: \hointr 0 \infty \to \hointr 0 \infty$ be a convex function.
Then for all positive measurable functions $g: X \to \R$, $g \in \map {\MM^+} \Sigma$:
- $\map V {\dfrac {\ds \int g \cdot f \rd \mu} {\ds \int f \rd \mu} } \le \dfrac {\ds \int \paren {V \circ g} \cdot f \rd \mu} {\ds\int f \rd \mu}$
where $\circ$ denotes composition, and $\cdot$ denotes pointwise multiplication.
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Concave Functions
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.
Also see
- Jensen's Inequality (Real Analysis), a similar result in the context of real analysis.
Source of Name
This entry was named for Johan Ludwig William Valdemar Jensen.
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $12.14$