By Ole A. Nielsen

This booklet describes integration and degree thought for readers drawn 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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**Extra info for An Introduction to Integration and Measure Theory**

**Sample text**

It is clear that co(/;x,r) < co(/;x,s) if 0 < r ^ s and hence that co (/;x ) = lim „_oQCo(/;x,r„) for any sequence (r„) of positive numbers such that lim„^oor„ = 0. 2). 2 A bounded function f defined on an interval is continous at a point X in its domain if and only if co{f;x) = 0. 3 I f f is a bounded function whose domain is a closed interval I then {x e I: co{f;x) ^ r} is a closed set for each positive number r. 2 N U LL S E T S Suppose that wis a nondecreasing function defined on the real line U.

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. 8 that every function in ^{u; a, b) is continuous at c.

Now, by hypothesis, there is an r > 0 such that co(/;x,r)< e, and then, in turn, there will be an integer m such that both x^ and y^ belong to (x —r, x + r), and hence such that If i x j - f i y j l < co(/; X,r) < e. But this is a contradiction. 8 Suppose that / is a bounded function and u a nondec reasing continuous function on an interval [a, b]. The theorem is true if u{b) = u(a) because in this case every bounded function on [a, b] belongs to ^{u; a, b) and every subset of [a, b] is w-null.