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). $\square$