Jensen’s inequality

Let $(𝜉, F, ℙ)$ be a probability model, $𝑋 : 𝜉 \to ℝ$ be a real random variable^[i.e. $ℙ$ -measurable function], and $𝜑 : ℝ \to ℝ$ be a Convex function. Then measure

𝜑 (\int 𝜉 𝑋 𝑑 ℙ) \leq \int 𝜉 𝜑 \circ 𝑋 𝑑 ℙ 𝜑 (𝔼 [𝑋]) \leq 𝔼 [𝜑 (𝑋)]

with equality iff $𝜑 (𝑋) = 𝑎 + 𝑏 𝑋$ .