Rademacher's teorem for euclidean measures

For every euclidean Radon measure $\mu$ we state an adapted version of Rademacher's theorem, which is, in a certain sense, optimal. We define a sort of fibre bundle (actually just a map $S$ that at each point $x\in\mathbb{R}^n$ associates a vector subspace $S(x)$ of $T_x\mathbb{R}^n$, possibly with non-costant dimension $k(x)$) such that every Lipschitz function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is differentiable at $x$, along the k(x)-plane $S(x)$, for $\mu$-a.e. $x$. We prove that $S$ is maximal in the following sense: if $T$ is a vector space such that $T(x)\not\in S(x)$ whenever $k(x)\neq n$, then there exists a Lipschitz function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ which doesn't admit any derivative in the direction $V(x)$ for $\mu$-a.e. $x$ satisfying $k(x)\neq n$.

Joint work with Giovanni Alberti.