Lectures on the Theory of Functions of Real Variables Volume 2
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Book 2
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Book 1
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Book 1
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Lectures on the Theory of Functions of Real Variables Volume 2

Lectures on the Theory of Functions of Real Variables Volume 2


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...For let S, J be a pair of complementary e/4 normal enclosures belonging to 21; let g, F be similar enclosures of SB. Let e = Z-(g, 1?), f-Dv% F). Then e e/2, f e/2, by 355, 2. Now (r) = Dv((, 5) is a normal metric enclosure of 35. Moreover its cells g which contain points of 35 and C(35) lie among the cells of e, f. Hence ae+T+ Thus by 357, 35 is measurable. 2. Let?(, S 6e measurable. Let Then For Hence Forlet o, -at-aI, o.-a.-a, .. For uniformity let us set at = 8. Then a = 2o-. As each Q is measurable 3 = 2 = lim(ai+ +o.)--00 = lira in. 361. Let 2Ij, S be measurable and their union 21 limited. If 3) = Dv 2l S 0, ft 18 measurable. For let 8 lie in the metric set 2ft; let (r) + D=m, an+4.=a- as usual. Now 3) denoting the points common to all the a, no point of D can lie in all of the a, hence it lies in some one or more of the An. Thus DAn. (1 On the other hand, a point of An lies in some Am, hence it does not lie in a . Hence it does not lie in 3). Thus it lies in D. Hence AnD. (2 From 1), 2) we have D= A As each An is measurable, so is D. Hence 3) is. 362. If aa is an enumerable set of measurable aggregates, their divisor 3) is measurable, and (c) = liman. B=ae For as usual let 2, An be the complements of 3), 8, with respect to some metric set 9ft. Then 2=M-j, 4.4.+1. Hence by 360, 3-Km A.. Let era=(emil, em2, e, 8...) + em, (3 then em is measurable. By this process the metric or measurable cell em falls into an enumerable set of non-overlapping measurable cells, as indicated in 3). If we suppose this decomposition to take place for each cell of (c), we shall say we have superimposed 8 on S. 364. (TF. H. Young. Let S be any complete set in limited SJ. Then = Max S. (1 F


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Product Details
  • ISBN-13: 9781231145890
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 96
  • Spine Width: 5 mm
  • Width: 189 mm
  • ISBN-10: 1231145897
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 186 gr


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