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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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).

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An introduction to Lebesgue integration and Fourier series by Howard J. Wilcox


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