If a function is Riemann integrable and Lebesgue integrable, the two integrals are the same?

Question

We know that if a proper Riemann integrable function is Lebesgue integrable, the two are equal. An improper Riemann integrable function is not necessarily Lebesgue integrable, like $\frac{\sin(x)}{x}$.

The question is that if a function $f$ (proper or improper)is Riemann Integrable and Lebesgue integrable, are the two integrals the same?

Answer 1

You already know the "proper" case. The "improper" case is a consequence of the Lebesgue dominated convergence theorem.