Jensen's inequality for integrals

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

This is a fundamental inequality for convex functions applied to the Lebesgue integral , often used with $f$ in L1 and $\varphi\circ f$ integrable. It can be viewed as a measure-theoretic extension of finite-dimensional convexity to averages taken with respect to a measure on a measure space .