ThmDex – An index of mathematical definitions, results, and conjectures.
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} \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}
Proofs
Proof 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} \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). $\square$