*Published Paper*

**Inserted:** 11 aug 2014

**Last Updated:** 28 may 2017

**Journal:** Geom. Funct. Anal. (GAFA)

**Volume:** 26

**Number:** 1

**Pages:** 1-66

**Year:** 2016

**Doi:** 10.1007/s00039-016-0354-y

**Abstract:**

For every finite measure $\mu$ on $\mathbb{R}^n$ we define a decomposability bundle $V(\mu,\cdot)$ related to the decompositions of $\mu$ in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on $\mathbb{R}^n$ is differentiable at $\mu$-a.e. $x$ with respect to the subspace $V(\mu,x)$, and prove that this differentiability result is optimal, in the sense that, following $[4]$, we can construct Lipschitz functions which are not differentiable at $\mu$-a.e. $x$ in any direction which is not in $V(\mu,x)$. As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) $k$-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class $C^1$ to Lipschitz maps.

**Keywords:**
Rademacher theorem, differentiability, Lipschitz functions, normal currents

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