By Ole A. Nielsen

ISBN-10: 0471595187

ISBN-13: 9780471595182

This ebook describes integration and degree conception 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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**Example text**

19 Show that if / is a Riemann integrable function on [a, ft], then f{x)dx = inf g{x)dx:geC[_a,b'] a n d ^ ^ / >. 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). (a) Show that v and w are nondecreasing, that v is right-continuous on (a, b), and that w is right-continuous on [a, b). (b) Show that ja / áw = / dv for all functions / eC[a, b].

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.

4 THE CH ARACTERIZA TION FOR CONTINUOUS INTEGRA TORS The promised characterization of Riemann-Stieltjes integrability in the case of continuous integrators is given by the following theorem. 8 Suppose that f is a bounded function and u a nondecreasing continuousfunction defined on a closed interval [a, h]. T h en f eJ{u\ a, b) if and only if the set o f points at which f is discontinuous is a u-null set. 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.

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