Resolution-limited measure and dimension

摘要

As extensions of Hausdorff measures and dimension, a new measure and associated dimension are introduced. Whereas the former concepts were intended to characterize sets as having fractional dimension, in the real world any set, even a computer- generated one, must have Hausdorff dimension which is exactly an integer. The new measure and dimension overcome this problem by explicitly limiting the resolution with which a set must be measured. The resulting development generally parallels Hausdorff's but makes it possible to attribute non-integer dimensions to real-world sets. Certain simplifications further enhance the practical usefulness of the concepts.