Exercise 1.11 (Riemann integrable functions are absolutely integrable)

Question

If $f: [a,b] ightarrow {\bf C}$ is a Riemann integrable function on a compact interval {[a,b]}, show that $f$ is also absolutely integrable, and that the Lebesgue integral $\int_{[a,b]} f\ dm$ (with $m$ Lebesgue measure restricted to {[a,b]}) coincides with the Riemann integral $\int_a^b f(x)\ dx$. Similarly if $f$ is Riemann integrable on a box $[a_1,b_1] 	imes \dots 	imes [a_n,b_n]$.

Elu Thingol · Accepted Answer

\begin{itemize}
\item On Lebesgue measurablity of the Riemann-integrable functions see \href{../an_introduction_to_measure_theory/exercise_1-3-9}{Exercise 1.3.9} from \href{../an_introduction_to_measure_theory}{Measure Theory}.
\item On compatibility of the Lebesgue integral and the Riemann integral, see \href{..an_introduction_to_measure_theory/exercise_1-3-17}{Exercise 1.3.17} from \href{../an_introduction_to_measure_theory}{Measure Theory}.
\end{itemize}

Exercise 1.11 (Riemann integrable functions are absolutely integrable)

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Exercise 1.11 (Riemann integrable functions are absolutely integrable)

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