Additive functions of intervals and Hausdorff measure
- 1 February 1946
- journal article
- research article
- Published by Cambridge University Press (CUP) in Mathematical Proceedings of the Cambridge Philosophical Society
- Vol. 42 (1) , 15-23
- https://doi.org/10.1017/s0305004100022684
Abstract
Consider bounded sets of points in a Euclidean space Rq 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 Rq, 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 {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We defineKeywords
This publication has 3 references indexed in Scilit:
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- Sets of k-Extent In n-Dimensional SpaceTransactions of the American Mathematical Society, 1933
- On linear sets of points of fractional dimensionMathematische Annalen, 1929