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THE HAUSDORFF MEASURE AND THE PACKING MEASURE ON A PERTURBED CANTOR SET

Meinershagen

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Abstract

Baek in [2] and [3] defines the covering measures $h^s$ and $Q^s$ for a perturbed Cantor set $F$. He shows that when $s$ is the dimension of the covering measures $h^{s\text{ }}$and $Q^s$ on $F$, then $s$ is the Hausdorff dimension and the packing measure dimension on $F$. In this paper, it is shown that for the perturbed Cantor set $F$, the Hausdorff measure is equal to the covering measure $h^s$ on $F$. Under more restrictions on the set $F$, the packing measure is equal to $2\cdot Q^s.$ Similar results are shown for the weakly convergent deranged Cantor set.

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Baek in [2] and [3] defines the covering measures $h^s$ and $Q^s$ for a perturbed Cantor set $F$. He shows that when $s$ is the dimension of the covering measures $h^{s\text{ }}$and $Q^s$ on $F$, then $s$ is the Hausdorff dimension and the packing measure dimension on $F$. In this paper, it is shown that for the perturbed Cantor set $F$, the Hausdorff measure is equal to the covering measure $h^s$ on $F$. Under more restrictions on the set $F$, the packing measure is equal to $2\cdot Q^s.$ Similar results are shown for the weakly convergent deranged Cantor set.

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Available abstract

Baek in [2] and [3] defines the covering measures $h^s$ and $Q^s$ for a perturbed Cantor set $F$. He shows that when $s$ is the dimension of the covering measures $h^{s\text{ }}$and $Q^s$ on $F$, then $s$ is the Hausdorff dimension and the packing measure dimension on $F$. In this paper, it is shown that for the perturbed Cantor set $F$, the Hausdorff measure is equal to the covering measure $h^s$ on $F$. Under more restrictions on the set $F$, the packing measure is equal to $2\cdot Q^s.$ Similar results are shown for the weakly convergent deranged Cantor set.

Key concepts: Mathematics, Hausdorff measure, Packing dimension, Cantor set, Measure (data warehouse), Cantor function, Hausdorff dimension, Outer measure

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