What is an example of lower semicontinuous functions not satisfying this?

Question

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$.

Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$.

Then, does there exist a lower semicontinuous function $U:X\rightarrow \mathbb{R}$ such that $U=u, \mu$-a.e.?