Let $X$ be a set with a symmetric kernel $d$ (not necessarily a distance).
The space $(X,d)$ is said to have the weak (resp. strong) covering
property of degree $\leq m$ [briefly \textbf{prf}$(m)$ (resp.
\textbf{prF}$(m)$)], if, for each family $\mathcal{B}$ of closed balls
of $(X,d)$ with radii in a decreasing sequence (resp. with bounded radii),
there is a subfamily, covering the center of each element of
$\mathcal{B}$, and of order $\leq m$ (resp. spliting into $m$ disjoint
families). Since Besicovitch, covering properties are known to be the main
tool for proving derivation theorems for any pair of measures on $(X,d)$.
Assuming that any ball for $d$ belongs to the Baire
$\sigma$-algebra for $d$, we show that the \textbf{prf} implies an almost
sure derivation theorem. This implication was stated by D. Preiss
when $(X,d)$ is a complete separable metric space. With stronger
measurability hypothesis (to be stated later in this paper), we
show that the \textbf{prf} restricted to balls with constant
radius implies a derivation theorem with convergence in measure.
We show easily that an equivalent to the \textbf{prf}$(m+1)$ (resp.
to the \textbf{prf}$(m+1)$ restricted to balls with constant radius)
is that the Nagata-dimension (resp. the De Groot-dimension) of
$(X,d)$ is $\leq m$. These two dimensions (see J.I. Nagata) are not
lesser than the topological dimension; for $\mathbb{R}^n$ with
any given norm ($n > 1$), they are $> n$. For spaces with
nonnegative curvature $\geq 0$ (for example for $\mathbb{R}^n$
with any given norm), we express these dimensions as the cardinality
of a net; in these spaces, we give a similar upper bound for the
degree of the \textbf{prF} (generalizing a result of Furedi and Loeb
for $\mathbb{R}^n$) and try to obtain the exact degree in
$\mathbb{R}$ and $\mathbb{R}^2$.