{"product_id":"9783032231420","title":"Convergence in Measure and in Category","description":"\u003ch3\u003eLecture Notes in Mathematics\u003c\/h3\u003e\u003ch1\u003eConvergence in Measure and in Category\u003c\/h1\u003e\u003ch3\u003eWładysław Wilczyński\u003c\/h3\u003e\u003cdiv\u003e\u003cb\u003eMathematics \/ Mathematical Analysis\u003c\/b\u003e\u003c\/div\u003e\u003cbr\u003e\u003cdiv\u003e\n\u003cp class=\"MsoNormal\" style=\"mso-margin-top-alt: auto; margin-bottom: .0001pt; line-height: normal;\"\u003e\u003cspan lang=\"EN-US\" style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003eThis book exhibits a vast gamut of similarities and differences between measure and (Baire) category. An important similarity is the Sierpiński–Erdős duality theorem: assuming the Continuum Hypothesis, there exists a one-to-one mapping \u003cem\u003ef\u003c\/em\u003e of the real line \u003c\/span\u003e\u003cspan lang=\"EN-US\" style=\"font-family: 'Cambria Math',serif; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'Cambria Math'; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003eℝ\u003c\/span\u003e\u003cspan lang=\"EN-US\" style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003e onto itself such that \u003cem\u003ef\u003c\/em\u003e(\u003cem\u003eA\u003c\/em\u003e) is a nullset if and only if \u003cem\u003eA\u003c\/em\u003e is of the first category. Moreover, this mapping can be chosen such that \u003cem\u003ef\u003c\/em\u003e = \u003cem\u003ef\u003c\/em\u003e\u003csup\u003e-1\u003c\/sup\u003e.\u003cspan style=\"mso-spacerun: yes;\"\u003e  \u003c\/span\u003eAn equally important difference is E. Szpilrajn’s theorem: there does not exist a mapping \u003cem\u003ef\u003c\/em\u003e of the real line \u003c\/span\u003e\u003cspan lang=\"EN-US\" style=\"font-family: 'Cambria Math',serif; mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: 'Cambria Math'; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003eℝ\u003c\/span\u003e\u003cspan lang=\"EN-US\" style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003e onto itself such that \u003cem\u003ef\u003c\/em\u003e(\u003cem\u003eE\u003c\/em\u003e) is Lebesgue measurable if and only if \u003cem\u003eE\u003c\/em\u003e has the Baire property.\u003c\/span\u003e\u003cspan style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-language: EN-IN;\"\u003e \u003c\/span\u003e\u003c\/p\u003e\r\n\u003cp class=\"MsoNormal\" style=\"mso-margin-top-alt: auto; margin-bottom: .0001pt; line-height: normal;\"\u003e\u003cspan lang=\"EN-US\" style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-IN;\"\u003eMuch of the book is devoted to the study of various modes of convergence: convergence almost everywhere; convergence except on a set of first category; convergence in measure; and convergence in the category of sequences of real functions of a real variable. Here, convergence in a category is *-convergence with respect to convergence except on a set of first category, just as convergence in measure is *-convergence with respect to convergence almost everywhere.\u003c\/span\u003e\u003cspan style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-fareast-language: EN-IN;\"\u003e \u003c\/span\u003e\u003c\/p\u003e\r\n\u003cp class=\"MsoNormal\" style=\"mso-margin-top-alt: auto; mso-margin-bottom-alt: auto; line-height: normal;\"\u003e\u003cspan lang=\"EN-US\" style=\"mso-fareast-font-family: 'Times New Roman'; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US;\"\u003eThe main focus is on sequences of real functions defined on the unit interval. If possible, theorems are proved in the more general setting of an abstract measurable space equipped with a sigma ideal. Sequences of functions that are divergent in measure or in category are also studied. In particular, the possibility of improving, destroying or preserving the convergence is addressed. The book will be valuable for those interested in real analysis and the theory of sequences or series of measurable functions.\u003c\/span\u003e\u003c\/p\u003e\n\u003c\/div\u003e\u003cdiv\u003e\u003cp class=\"MsoNormal\"\u003e\u003cstrong style=\"mso-bidi-font-weight: normal;\"\u003e\u003cspan lang=\"EN-US\" style=\"mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US;\"\u003eWładysław Wilczyński\u003c\/span\u003e\u003c\/strong\u003e\u003cspan lang=\"EN-US\" style=\"mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin; mso-ansi-language: EN-US;\"\u003e studied mathematics at the University of Łódź, where he received his Ph.D. His major research interests include real analysis and its interrelations with topology. In 1989, he led a semester on real analysis at the Banach Center (Warsaw, Poland). He is also the author of  Density Topologies (Chapter 15) in the \u003cem\u003e\u003cspan style=\"font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-theme-font: minor-latin;\"\u003eHandbook of Measure Theory\u003c\/span\u003e\u003c\/em\u003e, North Holland, Elsevier (2002), in which he presents the classical density topology of Haupt and Pauc and its category counterpart.\u003c\/span\u003e\u003c\/p\u003e\u003c\/div\u003e\u003cbr\u003e\u003ctable\u003e\n\u003ctr\u003e\n\u003ctd\u003ePublication Date: \u003c\/td\u003e\n\u003ctd\u003e24 July 2026\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003ePublisher: \u003c\/td\u003e\n\u003ctd\u003eSpringer Nature Switzerland\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eImprint: \u003c\/td\u003e\n\u003ctd\u003eSpringer\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eISBN-13: \u003c\/td\u003e\n\u003ctd\u003e9783032231420\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003eFormat: \u003c\/td\u003e\n\u003ctd\u003ePaperback \/ softback\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr\u003e\n\u003ctd\u003ePage Count: \u003c\/td\u003e\n\u003ctd\u003e137\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003c\/table\u003e","brand":"Springer Nature Switzerland","offers":[{"title":"Default Title","offer_id":46539653120140,"sku":"9783032231420","price":62.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9783032231420.jpg?v=1780591526","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9783032231420","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}