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!


Question: Driving electric cars through large pools of water
Answer: The answer to this question is what was put into its design at the drawing board stage. Encapsulation and water proofing of components susceptable to being submerged in water being the main consideration. As an electric vehicle construction is usually decided by its weight, less weight means less drain on the battery means greater distance between charges, encapsulation would not be a consideration. So I would be reluctant to drive through puddles higher then the hubs of the wheel at any sort of speed.