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}

Since $a \leq f \leq b$ almost everywhere, we may apply results

(i)	R1514: Isotonicity of signed basic integral
(ii)	R1499: Real-linearity of signed basic integral
(iii)	R1219: Simple integral is compatible with measure

to conclude \begin{equation} \int_E f \, d \mu \geq \int_E a \, d \mu = a \int_E \, d \mu = a \mu(E) \end{equation} and \begin{equation} \int_E f \, d \mu \leq \int_E b \, d \mu = b \int_E \, d \mu = b \mu(E) \end{equation} The latter two results along with result R1902: Signed basic integral of almost everywhere equal functions also establish claims (2) and (3). $\square$