Extend a measure to a Radon measure

Question

A Radon measure $\mu:\mathcal{P}(\mathbb{R}^n)\to [0,\infty]$ is an outer measure such that

1. Each Borel set $E\in\mathcal{B}(\mathbb{R^n})$ is $\mu$-measurable;
2. For every $F\subset \mathbb{R}^n$, there exists $E\in \mathcal{B}(\mathbb{R^n})$ such that $\mu(E)=\mu(F)$;
3. $\mu(K)<\infty$ if $K$ is compact.

Now let $\mu$ be a Radon measure, and let $u:\mathbb{R^n}\to \mathbb [0,\infty)$ be $\mu$-measurable. Define the set function $u\mu:\mathcal{B}(\mathbb{R^n}) \to [0,\infty]$ by $$u\mu(E)=\int_E ud \mu.$$ Then $u\mu$ is a Borel measure defined on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R^n})$.

I know that we can extend $u\mu$ to be an outer measure. My question is, is it possible to extend it to be a Radon measure?