Theorem (Jensen’s Inequality)

Let $f\in\mathscr{L}^{1}(\mathbb{R},\mathcal{B}(\mathbb{R}),\mu)$, let $\varphi:\mathbb{R}\to \mathbb{R}$ convex (implying continuity, implying measurability) and assume $\varphi\circ f\in\mathscr{L}^{1}(\mathbb{R},\mathcal{B}(\mathbb{R}),\mu)$. Also, $\mu(\mathbb{R})=1$ then $$\varphi\left( \int\limits _{\mathbb{R}}f \, d\mu \right)\le \int\limits _{\mathbb{R}}(\varphi\circ f) \, d\mu$$