I hope you can help me with a question on Lebesgue and Riemann integrals. The question is:
Is the Riemann integral an absolutely convergent integral as Lebesgue integral, that is, if $f$ is Riemann integrable then will be $\vert f \vert$ also integrable?
If it is not true, that is, if there is $f$ that is Riemann integrable but on the other hand $\vert f \vert$ is not Riemann integrable then, in this case, would be $f$ Lebesgue integrable? And in this case if $f$ is Lebesgue integrable, despite of $\vert f \vert$ is not Riemann integrable, $\vert f \vert$ would be Lebesgue integrable?
Thanks in advance.