# Geometric Integration Theory by Krantz S.G., Parks H.R.

By Krantz S.G., Parks H.R.

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Let Ak,1, Ak,2, . . be the subsequence of Ak−1,1 , Ak−1,2, . . 24) is true. Observe that Ak,i ⊂ Ck holds for sufficiently large i, else there is another subcube which could be added to the maximal collection while maintaining admissibility. 5. HAUSDORFF DISTANCE 35 and claim that C is the limit in the Hausdorff distance of Ak,k as k → ∞. Of course C is nonempty by the finite intersection property. Let > 0 be given. Clearly we can find an index k0 such that Ck0 ⊂ {x : dist(x, C) < }. There is a number i0 such that for i ≥ i0 we have Ak0 ,i ⊂ Ck0 ⊂ {x : dist(x, C) < }.

We let N denote the set of all finite sequences of positive integers and we let N denote the set of all infinite sequences of positive integers, so N = { (n1 , n2, . . , nk ) : k ∈ N+ , ni ∈ N+ for i = 1, 2, . . , k } , N = { (n1 , n2, . ) : ni ∈ N+ for i = 1, 2, . . } . 6 Let M be a collection of subsets of a set X. ,nk ∈ M associated with every finite sequence of positive integers. We can represent this relation as a function ν : N → M defined by ν (n1, n2 , . . ,nk . Such a function ν is called a determining system in M.

Let S ⊆ RN be a Suslin set, and let ν be a determining system in K such that S = N (ν). Since any finite intersection of compact sets is compact, we ν see that the determining system (n1 , n2, . . ,nk has the same 48 BASICS ν nucleus as the determining system (n1 , n2, . . ,nk = Kn1 ∩ Kn1 ,n2 ∩ . . ,nk . ) and so we see that f (S) is a Suslin set in RM . Measurability of Suslin Sets In order to prove that the Suslin sets are measurable, we need to introduce some additional structures similar to the nucleus of a determining system.