In the course I'm taking, we are already done with Lebesgue integration on R, and while we have proven that for continuous fonctions, the Riemann integral and the Lebesgue integral give the same output, we have not investigated further the correspondance btw the two. So I have some questions... i. Is the class of (absolutely) Riemann integrable function a subset of Lebesgue integrable ones? Or are there functions that are (absolutely) Riemann-integrable but whole Lebesgue integral diverge? ii. For those Riemann-integrable functions that are also Lebesgue integrable, are the Riemann and Lebesgue integral equal? I would also be interested to know what were the historical motivations for creating this new theory of integration, and also, what are its use today?