Additive functions of intervals and Hausdorff measure

Authors: P. A. P. Morán - Mathematical Proceedings of the Cambridge Philosophical Society 1946 cited by 404

Consider bounded sets of points in a Euclidean space R q of q dimensions. Let h(t) be a continuous increasing function, positive for t >0, and such that h (0) = 0. Then the Hausdroff measure h–mE of a set E in R q , relative to the function h(t) , is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets { U i } (open or closed) of diameters d i less than or equal to ε. Write h–m ε E = greatest lower bound for any such sequence { U i }. Then h–m ε E is non-decreasing as ε tends to zero. We define