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Let $U \subset \mathbb R^3$ be an open, bounded and connected set with a $C^2-$regular boundary $\Gamma$. If

  • $f \in \mathcal M_{+}(\Gamma)$ i.e belongs to the space of non negative, Radon measures on $\Gamma$ with finite mass
  • $\phi \in C^2(\Gamma)$ is an arbitary test function
  • $g \in L^{\infty}(\Gamma)$ then for constants $a,b \gt 0$ the following:

$\int_{\Gamma} af\phi = b\int_{\Gamma} \phi g+ \int_{\Gamma} \phi \int_{\Gamma} f$

implies that $af$ is an absolutely continuous measure.

I'm having a really hard time understanding the above implication. I know the definition of absolutely continuity for measures:

Let $\mu$ and $\nu$ be two measure on a $\sigma-$algebra $\mathcal B$ of subsets of $X$.Recall that $\nu$ is absolutely continuous with respect to $\mu$ if $\nu(A)=0$ for any $A \in \mathcal B$ such that $\mu(A)=0$.

Since there is no reference with respect to which measure $af$ is absolute continuous, I suppose it's with respect to the Lebesgue measure. However I can not proceed with this proof.

Any help or hint is appreciated. Thanks a lot in advance!

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