Extend a measure to a Radon measure

Mathematics Asked by inbrevi on January 1, 2022

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

1. Each Borel set $$Einmathcal{B}(mathbb{R^n})$$ is $$mu$$-measurable;
2. For every $$Fsubset mathbb{R}^n$$, there exists $$Ein mathcal{B}(mathbb{R^n})$$ such that $$mu(E)=mu(F)$$;
3. $$mu(K) if $$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 $$umu:mathcal{B}(mathbb{R^n}) to [0,infty]$$ by
$$umu(E)=int_E ud mu.$$ Then $$umu$$ is a Borel measure defined on the Borel $$sigma$$-algebra $$mathcal{B}(mathbb{R^n})$$.

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

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