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By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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E. point of A. This in fact depends upon the rectifiability of A, compare Theorem 5 in Sec. 4 Remark 1. Consider the measures 1-tk and 7{1, k _< n, in R n. Although they can be very different, it is easily seen that 7dk (A) = 0 if and only if Hk 00 (A) = 0. e. x E E, and one can show that, if Q, Qv v = 1, 2.... are compact se and if every open set A D Q contains Q for v sufficiently large, then 7{k (Q) > lim sup 7{1 (Qv) . v-oo Notice that the last inequality is in general false for the Hausdorff measure 7-Ck.

E. in X 1. e. x E X and we may take Z = {x E X I Dµv(x) = +oo} . (iv) of Theorem 3 hold if X = Rn. The proof of Theorem 3 is based on the following lemma whose proof uses in turn Vitali's property. In view of Theorem 2 (ii), the claims (i) Lemma 3. Let A be any subset of X. (i) If Dµv(x) < a in A, then v(A) < ay(A) (ii) If D,,v(x) > a in A, then v(A) > ap(A) A simple consequence of Theorem 3 is the following theorem to which we shall return in Sec. 1. Theorem 4 (Lebesgue-Besicovitch's differentiation theorem).

3 Hausdorff Measures rn/2 13 n=1,2,... (n/2)I'(n/2) where F(t) is Euler's gamma function 00 j xt-1 exp(-x) dx , we may and do take as ws WS :_ (s/2)F(s/2) . Fig. 1. k (E) - 2, f' (E) - 4. Definition 1. Let X be a metric space. i is then defined by 7-ls(A) (2) Observe that 71 bola(A), AC X > 7-1o2 for 61 < b2i thus 7-1 is well defined and 'H'(A) = supW (A) . 5>0 Notice that it is essential to use the approximating measures 7-lb in order to define H'. The naive definition ECUCk} 7ls(E) k k 1. General Measure Theory 14 in fact would have very unpleasant features.

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