Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON CANTOR SETS AND PACKING MEASURES

  • WEI, CHUN (DEPARTMENT OF MATHEMATICS SOUTH CHINA UNIVERSITY OF TECHNOLOGY) ;
  • WEN, SHENG-YOU (DEPARTMENT OF MATHEMATICS HUBEI UNIVERSITY)
  • Received : 2014.12.09
  • Published : 2015.09.30
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Abstract

For every doubling gauge g, we prove that there is a Cantor set of positive finite $H^g$-measure, $P^g$-measure, and $P^g_0$-premeasure. Also, we show that every compact metric space of infinite $P^g_0$-premeasure has a compact countable subset of infinite $P^g_0$-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 $\bar{F}=E{\cup}F$, and a doubling gauge g such that $E{\cup}F$ has different positive finite $P^g$-measure and $P^g_0$-premeasure.

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