Lebsesgue integrable functions as the closure of Riemann integrable functions

I've read that Lebesgue integrable functions can be thought of as the "completion" of Riemann integrable functions. I'm curious about the specific norm or metric in which this closure is taken.Additionally, is it true that the set of all Riemann integrable functions and their pointwise limits equals the set of all Lebesgue measurable functions on $\mathbb{R}$?