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

By Ole A. Nielsen

ISBN-10: 0471595187

ISBN-13: 9780471595182

This ebook describes integration and degree thought 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 imperative.

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Table Of Contents :
1. primary Integration Formulae, 2. Integration by way of Substitution - I, three. Integration through Substitution - II, four. Integration by means of elements, five. Integration by means of Partial Fractions, 6. sure crucial because the restrict of a Sum, 7. sure fundamental through the use of Indefinite imperative, eight. homes of yes Integrals, nine. quarter of Bounded areas utilizing yes Integrals, 10. Differential Equations, eleven. Homogeneous Differential Equations.

Extra info for An Introduction to Integration and Measure Theory

Sample text

8 Suppose that / is a bounded function and u a nondecreasing function on an interval [a, h]. Show that Exercises 35 (a) if there is a point in the interval [a, b) at which both / and и fail to be right-continuous, then / фJ^{u; a, b), and (b) if there is a point in the interval (a, b] at which both / and и fail to be left-continuous, then / a, b). 9 Let c be a point in an interval [a, b] and suppose that м is a nondecreasing function on [a, b] which is continuous at c and that / is a bounded function on [a, b] which is continuous at each point of [a, b] except possibly at the point c.

5 is just a special case of this theorem. Before turning to the proof of this theorem it might be useful to first state a corollary of it. Most of the assertions in this corollary were deduced in Chapter 3, but the point of stating them here again is to contrast the easy proof using the above theorem with the more difficult proof of the previous chapter. 9 Suppose that u is a nondecreasing continuous function defined on an interval [_a, b f {a) Any bounded function with only a countable number of points of discontinu­ ity, and hence any monotone function, on [a, b] belongs to J{u\ a, b).

These two calculations show that both (a) and (c) hold and that / = / du. To complete the proof it is sufficient to show that (a) implies (b), so suppose that (a) holds_and put / = / du. 2 with 5(/,w ;Pi) —5( / , w; ? 2) <£. 2 (b). 4 SO M E EX A M P LE S Before going to derive some of the properties of the Riemann-Stieltjes integral, it will be useful to present several examples. These examples will serve to point out several subtle features of the definitions of which it will be important to be aware.

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


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