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Jensen's inequality for integrals

A convexity inequality comparing a convex function of an average with the average of a convex function.

Jensen's inequality (integral form): Let (X,Σ,μ)(X,\Sigma,\mu) be a measure space with μ(X)=1\mu(X)=1. Let I⊆RI\subseteq\mathbb R be an interval, let f:X→If:X\to I be measurable and integrable, and let φ:I→R\varphi:I\to\mathbb R be convex. If φ∘f\varphi\circ f is integrable, then

φ(∫Xf dμ)≤∫Xφ(f) dμ.\varphi\Big(\int_X f\,d\mu\Big)\le \int_X \varphi(f)\,d\mu.

This is a fundamental inequality for applied to the , often used with ff in and φ∘f\varphi\circ f integrable. It can be viewed as a measure-theoretic extension of finite-dimensional convexity to averages taken with respect to a on a .