Question
Hi Folks!
Sorry I've been out of touch. You know, I've been trying to integrate myself (was area man, now volume man, plus constant). Anyway, I haven't had enough coffee, and I seem to proving contradictions in mathematics. Can anyone verify or debunk the following seemingly obvious statement? I have hopelessly confused myself.
Let C be the 1/3 Cantor set. Let s=log2/log3. Then Hausdorff s-measure(C)=1=Hausdorff s-content(C).
Right? Philosophy: For the Cantor set, the obvious cover is the most efficient. Of coursing proving this is a pain - see notes from Juha's course last semester.
Am I crazy?
Sorry I've been out of touch. You know, I've been trying to integrate myself (was area man, now volume man, plus constant). Anyway, I haven't had enough coffee, and I seem to proving contradictions in mathematics. Can anyone verify or debunk the following seemingly obvious statement? I have hopelessly confused myself.
Let C be the 1/3 Cantor set. Let s=log2/log3. Then Hausdorff s-measure(C)=1=Hausdorff s-content(C).
Right? Philosophy: For the Cantor set, the obvious cover is the most efficient. Of coursing proving this is a pain - see notes from Juha's course last semester.
Am I crazy?
posted by Kevin at 3:54 AM
1 Comments:
Right. Why is that H_{delta}^s does not depend on delta? Self-similarity. For example, H_1^s(C) = H_{1/3}^s(C) because a cover that gives H_1^s(C) can be rescaled by 1/3 and used twice. By induction, H_{(1/3)^n}^s(C) equals H_1^s(C), which is the same as the s-content of C. Take n->infty.
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