In particular if $f$ is as described, I want to show that the inproper integrals
$ \int_c^\infty f(x) dx$
are the same for Riemann and Lebesgue integral.
My idea is that, on compact intervals i.e. integrating from $c$ to $d>c$ it is true. See e.g. If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?.
Then the improper Riemann integral we take the limit of d to infinity.
But this is the same as taking the limit of a lebesgue integral, that is strictly increasing, so we can use monotone convergence.
Is this a correct argument, and is the statement that the improper integrals are equal true?