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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.