meas-square integrable function that is not integrable

Question

F the Borel sets, and u Lebesgue measure Show that there exists a mean-square integrable function on X that is not integrable if X=R

Answer 1

Let $f(x) = \frac {sin(x)}{x}$, then $\int_{\mathbb{R}} |f(x)|dx = \infty$, so $f(x)$ is not Lebesgue integrable. However, $\int_{\mathbb{R}} |f(x)|^2 dx \leq \infty$.