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. $mu(K)<infty$ 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 $umu:mathcal{B}(mathbb{R^n}) to [0,infty]$ 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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