Let f be bounded on $\mathbb{R}$. Illustrate an example that if f is Riemann integrable over $\mathbb{R}$, then it need not be Lebesgue integrable over $\mathbb{R}$. Determine a necessary condition under which Riemann integrability of f on $\mathbb{R}$ ensures the Lebesgue integrability.
I am following the book Measure Theory and Integration by G. de Barra and in the book, it has been discussed the equality of Riemann and Lebesgue integration over a finite interval. Can someone explain the relation between the two over the whole of $\mathbb{R}$.