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2015•Bulletin of the Korean Mathematical SocietyOpen access

ON CANTOR SETS AND PACKING MEASURES

Chun Wei, Shengyou Wen

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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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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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Available 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.

Key concepts: Mathematics, Cantor set, Countable set, Measure (data warehouse), Combinatorics, Null set, Metric space, Class (philosophy)

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