  Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from
below and $\pi:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for
any nonnegative function $f$ on $M$, the means of $f\o\pi$ on the geodesic
balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with
logarithmic volume estimates, this implies bounds on several topological
invariants (volume entropy, simplicial volume, first Betti number and
presentations of the fundamental group) in Ricci curvature $L^p$-bounded from
below.
