Jensen's Inequality

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space with $\mu$ a probability measure. Let $I$ be an interval that is an open subset of $(\mathbb{R},d_{\mathbb{R}})$ (Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces (iv)). Let $f\in\mathcal{L}^{1}(\mu)$ with $f(x)\in I$ for every $x\in X$, let $\varphi:I\to\mathbb{R}$ be convex on $I$, and suppose that the function $\varphi\circ f:X\to\mathbb{R}$, $x\mapsto\varphi(f(x))$, lies in $\mathcal{L}^{1}(\mu)$.

(i) The mean lies in $I$. $\int_{X}f\,d\mu\in I$.

(ii) Jensen's inequality.

$$\varphi\Bigl(\int_{X}f\,d\mu\Bigr)\le\int_{X}\varphi\circ f\,d\mu .$$