Entropy, approximation quantities and the asymptotics of the modulus of continuity

The paper deals with the approximation of bounded real functions f on a com pact metric space (X, d) by so-called controllable step functions in contin uation of [Ri/Ste]. These step functions are connected with controllable co verings, that are finite coverings of compact metric spaces by subsets whos e sizes fulfil a uniformity condition depending on the entropy numbers epsi lon(n)(X) Of the space X. We show that a strong form of local finiteness ho lds for these coverings on compact metric subspaces of IRm and Sm. This lea ds to a Bernstein type theorem if the space is of Finite convex deformation . In this case the corresponding approximation numbers (a) over cap(n) (f) have the same asymptotics as omega(f, epsilon n(X)) for f is an element of C(X). Finally, the results concerning functions f is an element of M(X) and f is an element of C(X) are transferred to operators with values in h(X) a nd C(X), respectively.