  We conclude our analysis of the linear response of charge transport in
lattice systems of free fermions subjected to a random potential by deriving
general mathematical properties of its conductivity at the macroscopic scale.
The present paper belongs to a succession of studies on Ohm and Joule's laws
from a thermodynamic viewpoint. We show, in particular, the existence and
finiteness of the conductivity measure $\mu _{\mathbf{\Sigma }}$ for
macroscopic scales. Then we prove that, similar to the conductivity measure
associated to Drude's model, $\mu _{\mathbf{\Sigma }}$ converges in the
weak$^{\ast } $-topology to the trivial measure in the case of perfect
insulators (strong disorder, complete localization), whereas in the limit of
perfect conductors (absence of disorder) it converges to an atomic measure
concentrated at frequency $\nu =0$. However, the AC--conductivity $\mu
_{\mathbf{\Sigma }}|_{\mathbb{R}\backslash \{0\}}$ does not vanish in general:
We show that $\mu _{\mathbf{\Sigma }}(\mathbb{R}\backslash \{0\})>0$, at least
for large temperatures and a certain regime of small disorder.
