By Richard Durrett

ISBN-10: 0534030653

ISBN-13: 9780534030650

This e-book will be of curiosity to scholars of arithmetic.

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Extra info for Brownian Motion and Martingales in Analysis (Wadsworth & Brooks Cole Mathematics Series)

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1953). SEC. 5 GENERAL THEORY OF THE INTEGRAL 31 Next we verify that the number Iq; is uniquely defined. ), because of the uniqueness of the integral in L+, and this implies (6) and hence (5). The integral just defined in the class L has the usual linearity properties. Let q; = f- g, q; 1 = / 1 - g 1 , where f, g, / 1 and g 1 belong to the class L+. , the integral of a sum equals the sum of the integrals. Moreover, if IX> 0, then f(1X({J) = f(IXj- 1Xg) = f(1Xj) - f(1Xg) = rx/f- IXfg = 1X(/f- Ig) = 1Xlq;.

Of the block B gives rise to a sequence of lower step functions ~ 1 (x), ... , ~0(x), ... and a sequence of upper step functions h1(x), ... , hq(x), ... Moreover, if every partition II«+t J8 THE RIEMANN INTEGRAL AND STEP FUNCTIONS CHAP. 1 is a refinement of its predecessor ITa, then the sequence of lower step functions ~ix) is nondecreasing and the sequence of upper step functions is nonincreasing. Assuming that d(I1a) -+ 0, we introduce the lower function j(x) = lim ~a(x) a-co and the upper function j(x) = lim ha(x), a-co where obviously f(x) < f(x) < f(x).

The converse is also true: If a function f(x) is the limit (in the sense of convergence almost everywhere) of some nondecreasing sequence kq{x) of step functions [not necessarily of the type ~q{x)], and at the same time the limit of a nonincreasing sequence of step functions lq{x), where kq{x) < f(x) < lq{x) everywhere, thenf(x) is Riemann integrable. To see this, let Ilq denote a partition of the block B into subblocks in which kq{x) is constant, and construct the corresponding functions ~q{x).

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Brownian Motion and Martingales in Analysis (Wadsworth & Brooks Cole Mathematics Series) by Richard Durrett


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