A Concise Introduction to the Theory of Integration - download pdf or read online

By Daniel W. Stroock

ISBN-10: 0817637591

ISBN-13: 9780817637590

ISBN-10: 3764337591

ISBN-13: 9783764337599

Designed for the full-time analyst, physicists, engineer, or economist, this publication makes an attempt to supply its readers with lots of the degree conception they are going to ever want. Given the alternative, the writer has constantly opted to increase the concrete instead of the summary facets of subject matters taken care of.

the key new characteristic of this 3rd version is the inclusion of a brand new bankruptcy during which the writer introduces the Fourier remodel. In that Hermite services play a principal position in his therapy of Parseval's identification and the inversion formulation, Stroock's strategy bears larger resemblance to that followed through Norbert Wiener than that utilized in most up-to-date introductory texts. A moment characteristic is that suggestions to all difficulties are supplied.

As a self-contained textual content, this ebook is superb for either self-study and the school room.

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Extra resources for A Concise Introduction to the Theory of Integration

Example text

D. nEIN Corollar. Die Menge In aller rationalen Zahlen ist abzählbar. § 9 Punktmengen 53 Beweis. Für jede natürliche Zahl n 2: I sind die Mengen An :={ ~: kEIN} und Bn := {- ~: kEIN} abzählbar. Nach Satz I ist deshalb auch Cn := {~: k E Z} abzählbar. Nun gilt CIl = = An UBn U Cn. Daher ist nach Satz I auch CIl abzählbar. n2:1 Satz 2. Die Menge IR aller reellen Zahlen ist überabzählbar. Beweis. Wir verwenden das sogenannte Cantorsche Diagonalverfahren. Es genügt zu zeigen, daß das Intervall )O,I[ nicht abzählbar ist.

D. =_1_=2 I 1- 2 -x . =_1_ =~ 1 3' 1 - (- 2:) 00 L ak und L bk zwei konvergente Reihen und A. E IR. ak) konvergent und k=O k=O k=O § 4 Folgen, Grenzwerte 25 es gilt 00 00 00 I (ak ± bk) = I 3k ± I bko k=O k=O k=O 00 00 IXak=XIak. k=O k=O Bemerkung. Für das Produkt zweier unendlicher Reihen gilt keine so einfache Formel, vgl. dazu § 8, Satz 3. n n Beweis. Sei c n := 1: ak und d n := 1: b k . Dann ist k=O k=O n n n I (ak + bk) = I 3k + I bk = Cn + d n k=O k=O k=O Nach Satz 3 gilt . 00 00 00 " (ak + b k ) = lim (cn + dn) = lim Cn + lim dn = I ak + I bk· ~ 0+00 0+00 0+00 k=O k=O k=O Die anderen Formeln werden analog auf die Corollare 1 und 2 von Satz 4 zurückgeführt.

Jeder b-adische Bruch stellt eine Cauchy-Folge dar, konvergiert also gegen eine reelle Zahl. 00 Beweis. Es genügt, einen nicht-negativen b-adischen Bruch L an b - n zu n=-k betrachten. L n Für n ~ -k sei xn := avb- v . v=-k Wir haben zu zeigen, daß die Folge (xn)n ~ -k eine Cauchy-Folge ist. Sei n ~ m ~ - k. Dann gilt IXn -xml = n v=m+1 < (b - l)b- m -I - L n _ _1__ 1-b- 1 v=m+1 =b- m < Damit ist die Behauptung bewiesen. L n-m-l (b-1)b- v :S;(b-l)b- m - 1 v=O € ' falls m hinreichend groß. b- v 30 § 5 Das Vollständigkeitsaxiom Satz 3.

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A Concise Introduction to the Theory of Integration by Daniel W. Stroock

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