Jensen's Inequality (Measure Theory)/Convex Functions
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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 $g: X \to \R$, $g \in \map {\MM^+} \Sigma$:
- $\map V {\dfrac {\ds \int g \cdot f \rd \mu} {\ds \int f \rd \mu} } \le \dfrac {\ds \int \paren {V \circ g} \cdot f \rd \mu} {\ds\int f \rd \mu}$
where $\circ$ denotes composition, and $\cdot$ denotes pointwise multiplication.
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Proof
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Let $\d \map \nu x := \dfrac {\map f x} {\ds \int \map f s \rd \map \mu s} \rd \map \mu x$ be a probability measure.
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Let $\ds x_0 := \int \map g s \rd \map \nu s$.
Then by convexity there exists constants $a, b$ such that:
\(\ds \map V {x_0}\) | \(=\) | \(\ds a x_0 + b\) | ||||||||||||
\(\ds \forall x \in \R_{\ge 0}: \, \) | \(\ds \map V x\) | \(\ge\) | \(\ds a x + b\) |
In other words, there is a tangent line at $\tuple {x_0, V_0}$ that falls below the graph of $V$.
Therefore:
\(\ds \map V {\map g s}\) | \(\ge\) | \(\ds a \map g s + b\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \map V {\map g s} \rd \map \nu s\) | \(\ge\) | \(\ds a \int \map g s \rd \map \nu s + b\) | Integration with respect to $\map \nu s$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \map V {x_0}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map V {\int \map g s \rd \map \nu s}\) |
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