I have that $\mu$ is a Borel probability measure on $\mathbb{R}$. When can I write:
$$ \lim_{u\rightarrow 0} \int_{\mathbb{R}} \frac{\sin(\frac{ux}{2})}{(\frac{ux}{2})} x^{2} d\mu(x) = \int_{\mathbb{R}} \lim_{u\rightarrow 0} \frac{\sin(\frac{ux}{2})}{(\frac{ux}{2})} x^{2} d\mu(x) $$
I don't see a way to use Dominated Convergence. Would Uniform Convergence help me?