By Ole A. Nielsen

ISBN-10: 0471595187

ISBN-13: 9780471595182

This e-book describes integration and degree concept for readers attracted to research, engineering, and economics. It supplies a scientific account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes degree from the Lebesgue-Stieltjes critical.

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**Sample text**

But then ^ U (X )c -« 5 ,X t + \jeJ J \k =0 <5) and jeJ k= 1 and since s is arbitrary, this shows that is a w-null set. Now suppose that, conversely, is a w-null set for each r > 0, and fix an £ > 0. Let M = sup^^^^j, / (x) and m = inf^^^c^b / W? r be a positive number satisfying M —m + u{b) — u(a) < sir. , are pairwise disjoint by successively replacing two intervals that are not disjoint by their union until the resulting intervals are pairwise disjoint. 10) there is a > 0 such that l/(x) —f{y)\ < r for all x,yelXj_^,Xj'] with |x —yl < öj.

If x^ _ ^ < c < Xj for some integer j, then S{f, u; P) = sup{/(x): ^ ^ x ^ x^} < /(c) + c/3 and S(/, u; P) = inf(/(x): Xj_ i < x < x / > /(c) - c/3, and therefore S (/, u; P ) - S ( / , m;P )< 8 . +1} Properties of Riemann-Stieltjes Integrals 27 and S{f, u;P) = a inf{/(x): ^ x < c} + (1 - a) inf{/(x): c ^ x ^ x^^ i } ^ a(/(c) - 8/3) + (1 - a)(/(c) - 8/3), and therefore S {f,u ;P )-S { f,u ;P )< 8 . 4 imply that / eJ^{u;a,b) and that ¡ lfd u = f (c) and that, moreover, given an г > 0, there is a <5> 0 such that if P is any partition of [0,1] with mesh(P) < <5, then S{f, P) — S{f, P)< e and \^ lfd u — S \< e for any Riemann-Stieltjes sum of / with respect to u and P.

Next, remove the open middle third from each of these two closed intervals, leaving four closed intervals each of length 1/9. Now repeat the process, removing the open middle third from each of these four closed intervals, leaving eight closed intervals each of length 1/27. The Cantor set C is the set that one is left with after continuing this process recursively for neN. 13). The Cantor function is a function that is closely related to the Cantor set and that is equally useful in constructing pathological examples in analysis.

### An Introduction to Integration and Measure Theory by Ole A. Nielsen

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