structural:expansion, quality:correction
Probabilistic form

The same result can be equivalently stated in a probability theory setting, by a simple change of notation. Let (Ω, F, P) {\displaystyle (\Omega,{\mathfrak {F}},\operatorname {P})} be a probability space, X an integrable real-valued random variable and φ {\displaystyle \varphi } a convex function. Then:

In this probability setting, the measure μ is intended as a probability P {\displaystyle \operatorname {P} }, the integral with respect to μ as an expected value E {\displaystyle \operatorname {E} }, and the function f {\displaystyle f} as a random variable X.

Note that the equality holds if and only if φ {\displaystyle \varphi } is a linear function on some convex set A {\displaystyle A} such that P (X ∈ A) = 1 {\displaystyle \mathrm {P} (X\in A)=1} (which follows by inspecting the measure-theoretical proof below).