On the comparison of measure functions

Abstract

Suppose that f and g are measure functions and that µf and µg are the corresponding Hausdorff measures. We are interested in how relationships between f and g are reflected in relationships between µf and µg and vice versa. For example, ifRogers ((8), p. 80) has shown that there exists a metric space Ω which has subsets A and B with µf(A) = µg(B) = 0 and, µf(B) = µg(A) = ∞. We are more interested in what happens when the metric space in question is the real line. To get the above result we need to assume that both f and g are starshaped. This is just about a best possible result since if we assume only that f is a power of t the conclusion fails. Further-more we show that there exist measure functions f and g satisfying (1) such that µf and, µg are identical. We also consider related questions: when are, µf and, µg equivalent measures and when are they identical?The proofs depend on the construction in Theorem 3.1 of a Cantor set having pre-scribed measure properties. Although the construction is not difficult it turns out to be quite useful to appeal to the existence of such a set. We illustrate this remark by giving a short proof of a known theorem on cartesian product sets. We also make use of these ideas in section 5 where we discuss some properties of a class of net meastues and give a partial answer to a problem posed by Billingsley.