I was reading a paper when I found this ($\partial \Omega$ refers to the boundary of $\Omega$ and $\nabla$ to the gradient operator,$\nabla f = (\partial_{i}f)_{i} $ ).
Let $\Omega \subset \mathbb{R}^{n}$ an open set and define $$\mu := \nabla 1 _{\Omega}$$ If $\mu$ is a locally finite $\mathbb{R}^{n}$-valued measure, then it follows from the Radon-Nikodym theorem that $\mu = -\nu \sigma$, where $\sigma$ is a locally finite positive measure,supported on $\partial \Omega$ and $\nu \in L^{\infty}(\partial \Omega, \sigma)$ is an $\mathbb{R}^{n}$-valued function,satisfying $|\nu(x)|=1$, $\sigma$ - almost everywhere.
I was wondering how can we define a measure like this, as $\nabla 1_{\Omega}$ is constant in $\Omega$ and I don't know how to define its partial drivatives in the boundary so that they define a measure there. On the other hand, the second part doesn't fit with the version of Radon-Nikodym theorem that I know and I can't see the conection.
Any help would be apprecciated, thanks!
The gradient is to be understood as a distributional derivative. This is an operation which makes sense on any function that is locally integrable.
As for the Radon-Nikodym Theorem, write $\mu=(\mu_1,\ldots,\mu_n)$. When restricted to a compact set, the components are finite, real measures, so it makes sense to consider the positive, finite measure $$ |\mu|=\mu_1^++\mu_1^-+\ldots+\mu_n^++\mu_n^-, $$ where $\mu_j=\mu_j^+-\mu_j^-$ is the Hahn Decomposition of $\mu_j$. It follows that $\mu_j\ll |\mu|$ for all $j$. By the Radon-Nikodym Theorem, there is a function $\nu=(\nu_1,\ldots,\nu_n)$ such that $\mu=\nu \sigma$, where $\sigma=|\mu|$.