Hausdorff, Packing and Capacity Dimensions Page: 13
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13
Comparing Hausdorff Packing Measures
The following lemma and theorem are due to Taylor and
Tricot [41, pp. 683-691]. The proofs presented here are
different from the proofs given in [41].
Lemma 2.4. For any E in B and any measure function,
(23) ^(E) < c^-P^(E)
with C(p the smoothness constant for <p. Therefore, if P^(E)
< oo, then "^(E) < oo.
Proof. From theorem 2.3, we see that, for 6 > 0, it is
possible to construct a Sg(E)-packing of E, Q, such that
|r*: R. € is a Sg^-cover of E. Therefore,
(24) ^c/ X c(dia" ®r(x)) * c»-I>«(E)-
Br(x)€ff
Let <5 go to zero. ■
Theorem 2.5. For all E 6 B, ^(E) < P^(E) .
Proof. Assume P^(E) < oo. For e > 0, there is a p > 0
such that
(25) ^(E) < *£(E) + e and PJJ(E) < P<"(E) + <=.
Let
(26) ^ = |Br(x): x € E, 2r < j-pJ.
For some set Q which is either an initial segment of IN or is
IN, there is a ^-packing of E,
(27) *={H.|i€Q},
such that if n is a positive integer and is less than the
cardinality of Q, then
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Spear, Donald W. Hausdorff, Packing and Capacity Dimensions, dissertation, August 1989; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc330990/m1/18/: accessed April 20, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .