About the H - Measure of a Set . Ii .

In the papers [B1] and [B2] we have established some conditions for the finitude of the Hausdorff h-measure of some set. Now, we shall determine a better minorant for this measure. 1 Definitions We denote by R the euclidean n dimensional space and by d(E) the diameter of a set E ⊂ R. Definition 1 If r0 > 0 is a fixed number, a continuous function h(r), defined on [0, r0) , nondecreasing and such that limr→0h(r) = 0 is called a measure function. If E ⊂ R is a bounded set and δ ∈ R+, the Hausdorff h-measure of E is defined by: Hh(E) = lim δ→0 inf ∑ i h(ρi), inf being considered over all coverings of E with a countable number of spheres of radius ρi ≤ δ. Definition 2 f : D(⊂ R) → R is a δ class Lipschitz function if |f(x + α) − f(x)| ≤ M |α| δ , x ∈ D,α ∈ R, x + α ∈ D,M > 0. Received: June, 2001