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} \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

(1)	\begin{equation} a \mu(E) \leq \int_E f \, d \mu \leq b \mu(E) \end{equation}
(2)	\begin{equation} \int_E f \, d \mu = a \quad \iff \quad f \overset{a.e.}{=} a \end{equation}
(3)	\begin{equation} \int_E f \, d \mu = b \quad \iff \quad f \overset{a.e.}{=} b \end{equation}