Some properties of packing measure with doubling gauge

Sheng-You Wen; Zhi-Ying Wen

Studia Mathematica (2004)

  • Volume: 165, Issue: 2, page 125-134
  • ISSN: 0039-3223

Abstract

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Let g be a doubling gauge. We consider the packing measure g and the packing premeasure ₀ g in a metric space X. We first show that if ₀ g ( X ) is finite, then as a function of X, ₀ g has a kind of “outer regularity”. Then we prove that if X is complete separable, then λ s u p ₀ g ( F ) ≤ g ( B ) ≤ s u p ₀ g ( F ) for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite ₀ g -premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge function, there is a compact metric space of finite positive packing measure.

How to cite

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Sheng-You Wen, and Zhi-Ying Wen. "Some properties of packing measure with doubling gauge." Studia Mathematica 165.2 (2004): 125-134. <http://eudml.org/doc/284678>.

@article{Sheng2004,
abstract = {Let g be a doubling gauge. We consider the packing measure $^\{g\}$ and the packing premeasure $₀^\{g\}$ in a metric space X. We first show that if $₀^\{g\}(X)$ is finite, then as a function of X, $₀^\{g\}$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $λsup₀^\{g\}(F) ≤ ^\{g\}(B) ≤ sup₀^\{g\}(F)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite $₀^\{g\}$-premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge function, there is a compact metric space of finite positive packing measure.},
author = {Sheng-You Wen, Zhi-Ying Wen},
journal = {Studia Mathematica},
keywords = {packing measure; premeasure; gauge function; doubling condition},
language = {eng},
number = {2},
pages = {125-134},
title = {Some properties of packing measure with doubling gauge},
url = {http://eudml.org/doc/284678},
volume = {165},
year = {2004},
}

TY - JOUR
AU - Sheng-You Wen
AU - Zhi-Ying Wen
TI - Some properties of packing measure with doubling gauge
JO - Studia Mathematica
PY - 2004
VL - 165
IS - 2
SP - 125
EP - 134
AB - Let g be a doubling gauge. We consider the packing measure $^{g}$ and the packing premeasure $₀^{g}$ in a metric space X. We first show that if $₀^{g}(X)$ is finite, then as a function of X, $₀^{g}$ has a kind of “outer regularity”. Then we prove that if X is complete separable, then $λsup₀^{g}(F) ≤ ^{g}(B) ≤ sup₀^{g}(F)$ for every Borel subset B of X, where the supremum is taken over all compact subsets of B having finite $₀^{g}$-premeasure, and λ is a positive number depending only on the doubling gauge g. As an application, we show that for every doubling gauge function, there is a compact metric space of finite positive packing measure.
LA - eng
KW - packing measure; premeasure; gauge function; doubling condition
UR - http://eudml.org/doc/284678
ER -

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