When asked what are the advantages of the Lebesgue integral over the Riemann integral, a common answer is that the space of Riemann integrable functions is not complete, whereas the space of Lebesgue integrable functions is complete.
However, I think the space of Riemann integrable functions on a closed interval is complete with respect to the uniform norm.
https://en.wikipedia.org/wiki/Uniform_convergence#To_integrability
What is the advantage of the Lebesgue integral over the Riemann integral in terms of completeness of the space of integrable functions?