ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation F10502 on R4513:
Formulation 0
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}