  In this paper we completely solve the problem of finding the (upper)
approximation order with respect to the Kolmogorov, Gel'fand, and linear widths
for the embedding of the Sobolev spaces $W^{\alpha,p}$ and $W_{0}^{\alpha,p}$
in the euclidian measure spaces $L_{\nu}^{q}$ for an arbitrary Borel
probability measure $\nu$ with support contained in the open $m$-dimensional
unit cube and for all possible choices of $1\leq p,q\leq\infty$. We will
determine the exact values for the various upper approximation orders in terms
of the $L^{q}$-spectrum of $\nu$ only and finally give sufficient conditions
imposed on the regularity of the $L^{q}$-spectrum for the approximation orders
to exist. We also elucidate some intrinsic connections between the concept of
approximation order and the fractal geometric notion of the upper and lower
Minkowski dimension of the support of $\nu$.
