By I. Madsen, B. Oliver

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Finally we come to the triangle inequality. Let S, T, U ∈ B. Let s ∈ S, t ∈ T, u ∈ U. Then we have |s − u| ≤ ⇓ dist(S, u) ≤ ⇓ dist(S, u) ≤ ⇓ dist(S, u) ≤ ⇓ dist(S, u) ≤ ⇓ dist(S, u) ≤ |s − t| + |t − u| |s − t| + |t − u| dist(S, t) + |t − u| HD (S, T ) + |t − u| HD (S, T ) + dist(T, u) HD (S, T ) + sup dist(T, u) u∈U ⇓ sup dist(S, u) ≤ HD (S, T ) + sup dist(T, u). u∈U u∈U By symmetry, we have sup dist(U, s) ≤ HD (U, T ) + sup dist(T, s) s∈S s∈S and thus max{ sup dist(S, u) , sup dist(U, s) } u∈U s∈S ≤ max{HD (S, T ) + sup dist(T, u) , HD (U, T ) + sup dist(T, s)}.

But it is worth stating separately as it is the basis for the theory of Hausdorff dimension. When F is the family of all closed balls in RN , and ζ1 as above, then the resulting measure ψ is called the m-dimensional spherical measure. 1. THE BASIC DEFINITION 57 denote this measure by S m . The same measure results if we use the family of all open balls. Of course it is immediate that Hm ≤ S m ≤ 2m · Hm . More precise comparisons are possible, and we shall explore these in due course. 2 A Measure Based on Parallelepipeds Let M > 0 be an integer and assume that M ≤ N , the dimension of the Euclidean space RN .

To see this claim, suppose it were not true. ,nk(n1 ) whenever ni ≤ hi for i = 2, 3, . . , k(n1 ). Setting K(1) = max{ k(1), k(2), . . 38). Arguing inductively, suppose we have selected positive integers n01 , n02 , . , n0s satisfying n01 ≤ h1 , n02 ≤ h2 , . , n0s ≤ hs , for every k with s + 1 ≤ k, there exist ns+1 , ns+2 , . . ,nk . 39) holds with s replaced by s +1. ,nk(ns+1 ) . whenever ni ≤ hi for i = s + 1, s + 2, . . , k(ns+1 ). Setting K(s + 1) = max{ k(1), k(2), . . 38). Thus there exists an infinite sequence n01 ≤ h1 , n02 ≤ h2, .