 By Howard J. Wilcox

ISBN-10: 0882756141

ISBN-13: 9780882756141

Undergraduate-level advent to Riemann fundamental, measurable units, measurable services, Lebesgue crucial, different subject matters. quite a few examples and routines.

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This booklet is an introductory textual content in sensible research. in contrast to many sleek remedies, it starts off with the actual and works its method to the extra common. From the reports: "This ebook is a superb textual content for a primary graduate direction in sensible research. .. .Many attention-grabbing and critical purposes are integrated.

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Additional info for An introduction to Lebesgue integration and Fourier series

Example text

Proof: (•) . 6. (�) . Let Pn be a partition of [-n,n] (on the y -axis) ob· tained by taking equal sub-intervals of length 1/n. ) 2 n2 Now let B; = { x E A IYi - 1 < f(x) < y,} , and let In = � Yt - 1 "XB i · i- 1 where Yt = - n • Simp • 1e. Then 1I"n IS Now given x0 E A , for each n big enough so that f(x0) E [-n,n] , Yt- 1 < f(x o) < Yi for some i = 1 ,2, • • • ,2n 2 • Thus x0 E Bi and 1/(x o) - /n(X o) l = lf(x o ) - Yt- 1 1 < lY; -Yt- 1 1 = 1/n. MEAS U R A B L E F U NCT I O NS So lim• /11(Xo) = f(xo) for all x0 E A .

7. x >. e. on a compact set K , then for every C K such that f is bounded on A , and m(K\A ) < e. e. e. (b) Prove or disprove the converse of (a). e. on a bounded measurable set A , then I is measurable. (By Exercise 20. 1 6(b), you may not use Proposition 1 8. e. " is an equivalence relation on the class of all functions on a bounded measurable set A . 2 that able if / is measurable. 20 Prove or disprove: if 1/1 is measurable, so is /. 21 Prove Corollary 1 8. 5 . 22 Show that f i s measurable on A if and only i f /2 is measurable and {x E A l fi 0} is measurable.

5 Example : (The Cantor Set) Let PROPERTI ES OF M EASU R A B LE SETS c, : ! � ,3----2/ ·3 35 ... ____ c. . 0 • 1/9 • 2/9 • • 1/3 2/3 • 7/9 • 8/9 • I That is, C,. 6 Definition : The Cantor Set ... C = nn= l C,. 7 Theorem : The Cantor Set is an uncountable measurable set with measure zero. Proof: Clearly C is closed since its complement is open . Thus C is measurable. ) =fm(C,. _ I) = f(fm(C,. , so by Corollary 1 0. 1 0 m(C} = lim C,. = lim (1.. 3 )" = 0. e. base three). (See Exercise 1 6. g. 2000 • • • here).