Prove that if $|f|$ is both Lebesgue integrable and Riemann integrable, then the integrals are equal.

Question

I started by proving that $f$ is Lebesgue integrable iff $|f|$ is, as follows: $f= f^+ - f^- $, and $|f|= f^+ + f^-$, where $f^+$ is when $f$ takes positive values, and similarly for $f^-$. And if $|f|$ is Riemann integrable, then that means $|f|$ is bounded, and so is $f$, which means $f$ is also Riemann integrable. Now how do I conclude that these 2 are equal?

Answer 1

Here’s a hint: the lower Darboux (Riemann) sums can be viewed as integrals of simple functions. Then you just need to refine your partition and use the Monotone Convergence Theorem.