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By Ole A. Nielsen

ISBN-10: 0471595187

ISBN-13: 9780471595182

This ebook describes integration and degree idea for readers attracted to research, engineering, and economics. It offers a scientific account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes degree from the Lebesgue-Stieltjes imperative.

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Extra info for An Introduction to Integration and Measure Theory

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Show that m°9 is a nondecreasing function on [c,ii], that / °deJ^(u°d;c,d), and that f du= \ f °9ii(w°9). (b) Suppose that 9 is strictly decreasing and continuous and satisfies 9(c) = b and 9(ii) = a. Show that —w°9 is a nondecreasing function on [c,ii], that / o9 gJ^( —M°9;c,. 20 Let Mbe a nondecreasing function on an interval [a, fc] and define two functions Vand w on [a, b] as follows: v{b) = w(b) = u(b\ v{a) = u{a), v(x) = w(x + ) for xe{a, b), and w(x) = u(x + ) for xe[a,b).

Let s be a given positive number. As u is nondecreasing and hence has a right limit at c there must be a number <5e(0, b — c) such that 0 ^ u{t) — u(s) < г/2 for all numbers s and t satisfying c < s ^ t ^ c + 5. 12(b)]. /=1 7=0 n-l ^ = ( l / 7i)w(c-l-a) — m(c) + Yj - w(Xj ) 7=1 1 " =~ E u{X j)-u{c). J^-iu{c + )-u{c)^ «c + S(

12 Define a function / on [0,1] by the formula /w = 0, 1/n, X is irrational or x = 0 X = min with m and n relatively prime positive integers. Show that / is Riemann integrable on [0,1] and determine the value of /(x)dx. 10) Suppose that м is a nondecreasing function on an interval [a, b] and that ( / J is a sequence of functions in a, b). Show that iflim„^oo/„ = / uniformly on [a, b] for some function / on [a, b],then f e J ( u ; a,b) and / „ du = /d n . 14 Suppose that / is a bounded function and и a nondecreasing function on an interval [a, b] such that / eJ^{u; a, b).

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


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