Is it possible that a set A has Hausdorff dimension d = <mrow class="MJX-TeXAt

hryggcx

hryggcx

Answered question

2022-07-10

Is it possible that a set A has Hausdorff dimension d = d i m H ( A ) ( 0 , ) but H d ( A ) ( 0 , )? In other words, positive and finite Hausdorff dimension but its Hausdorff measure is never positive and finite?

Answer & Explanation

grubijanebb

grubijanebb

Beginner2022-07-11Added 10 answers

An example is the real line R with its usual metric. Then
1 = dim H ( R ) , but H 1 ( R ) = +
An example of a subset A of R with
1 = dim H ( A ) , but H 1 ( A ) = 0 .

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