On Cantor sets and packing measures
Bull. Korean Math. Soc. 2015 Vol. 52, No. 5, 1737-1751
https://doi.org/10.4134/BKMS.2015.52.5.1737
Printed September 30, 2015
Chun Wei and Sheng-You Wen
South China University of Technology, Hubei University
Abstract : For every doubling gauge $g$, we prove that there is a Cantor set of positive finite $\mathcal{H}^g$-measure, $\Hp^g$-measure, and $\mathcal{P}_0^g$-premeasure. Also, we show that every compact metric space of infinite $\mathcal{P}_0^g$-premeasure has a compact countable subset of infinite $\mathcal{P}_0^g$-premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set $E$ in this class, there exists a countable set $F$, with $\overline{F}=E\cup F$, and a doubling gauge $g$ such that $E\cup F$ has different positive finite $\Hp^g$-measure and $\mathcal{P}_0^g$-premeasure.
Keywords : Cantor set, packing measure, premeasure, gauge function, doubling condition
MSC numbers : 28A78, 28A80
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