Are integrals in Riemann-Lebesgue theorem Riemann or Lebesgue

Question

Riemann-Lebesgue theorem says that if $f$ is Lebesgue-integrable on $\mathbb R$ that

\begin{equation} \lim_{n\to+\infty}\int_{-\infty}^{+\infty}f(x)\cos(nx)\,dx=0. \end{equation}

Are integrals $\int_{-\infty}^{+\infty}f(x)\cos(nx)\,dx$ Riemann improper integrals or Lebesgue integrals? If they are Riemann, why do they exist?

Answer 1

Only assumption on $f$ is : $f\in L^1$. Thus obviously the integral is Lebesgue integral.