Prove that if $f$ is Riemann integrable on $[0,1]$, then $f$ is measurable and it equals to Lebesgue integral.

Question

Prove that if $f$ is Riemann integrable on $[0,1]$, then $f$ is measurable and it's Riemann integral equals to Lebesgue integral on $[0,1]$.

I don't understand the two kinds of integrals pretty well. Where should I start? Any hints could be helpful.

Answer 1

Use the fact that for f to be Riemann-integrable, it must be a.e. continuous, and an a.e. continuous function is measurable. You also need restrictions on the interval of integration; I think you need compactness:

Measurability of almost everywhere continuous functions

For the agreement, you use some of the convergence theorems. See, e.g.

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?