ThmDex – An index of mathematical definitions, results, and conjectures.

Let $M = (X, \mathcal{F}, \mu)$ be a D1158: Measure space such that

(i)	$E \in \mathcal{F}$ is a D1109: Measurable set in $M$
(ii)	\begin{equation} 0 < \mu(E) < \infty \end{equation}
(iii)	$f : X \to [-\infty, \infty]$ is an D1921: Absolutely integrable function on $M$

Let $[a, b] \subseteq [-\infty, \infty]$ be a D4658: Closed basic interval such that

(i)	\begin{equation} a \overset{a.e.}{\leq} f \overset{a.e.}{\leq} b \end{equation}

Then \begin{equation} a \leq \frac{1}{\mu(E)} \int_E f \, d \mu \leq b \end{equation}