If f_n goes uniformly to 0 and its integral w.r.t the Leb. measure converges to 0, then its integral w.r.t a locally finite measure converges?

Assume $(f_n)_{n=1}^{\infty}$ is a sequence of functions defined on the real line such that $\lim_{n\to\infty}f_n = 0$ uniformly on $\mathbb{R}$, and assume that $\lim_{n\to\infty}\int_{\mathbb{R}}f_n(x)\mathrm{d}x=0$. Let $\mu$ be a locally finite measure on $\mathbb{R}$. Is it true that $\lim_{n\to\infty}\int_{\mathbb{R}}f_n(x)\mathrm{d}\mu(x) = 0$?