Here $\sigma$ and $\mu$ are such that $\int_U{f\,\text{div}\phi}= \int_U{\phi\,\sigma\, d\mu}$ for all open sets $V\subset\subset U$ and $\phi$ in $C_c^1(V, \mathbb{R}^n)$. But I am not sure what $[Df]$ and $\mu^i$ are supposed to be. $[Df]$ is supposed to equal $\sigma$ integrated with respect to the measure $\lVert Df \rVert$? So it's a vector? And is $\mu^i$ the i-th component of that vector? If so, then $\mu^i$ is not necessarily a measure, because it is signed. So how is it that we can use Lebesgue's decomposition theorem?