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) $$\mu(E) < \infty$$ (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) $$a \overset{a.e.}{\leq} f \overset{a.e.}{\leq} b$$
Then
 (1) $$a \mu(E) \leq \int_E f \, d \mu \leq b \mu(E)$$ (2) $$\int_E f \, d \mu = a \quad \iff \quad f \overset{a.e.}{=} a$$ (3) $$\int_E f \, d \mu = b \quad \iff \quad f \overset{a.e.}{=} b$$
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) $$\mu(E) < \infty$$ (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) $$a \overset{a.e.}{\leq} f \overset{a.e.}{\leq} b$$
Since $a \leq f \leq b$ almost everywhere, we may apply results
to conclude $$\int_E f \, d \mu \geq \int_E a \, d \mu = a \int_E \, d \mu = a \mu(E)$$ and $$\int_E f \, d \mu \leq \int_E b \, d \mu = b \int_E \, d \mu = b \mu(E)$$ The latter two results along with result R1902: Signed basic integral of almost everywhere equal functions also establish claims (2) and (3). $\square$