Proof verification: To show that a function is not Lebesgue integrable.
Choose $T$ such that $\int_0^{T}|g(t)|dt >0$. Note that $\int_1^{\infty}\int_0^{y} \frac {|g(t)|} y dt dy \geq \int_T^{\infty}\int_0^{y} \frac {|g(t)|} y dt dy$. Now use the fact that $\int_0^{y}|g(t)|dt \geq \int_0^{T}|g(t)|dt$ for $y >T$. Can you finish?