LEADER 03815nam 22005895 450 001 9910254074203321 005 20200703113330.0 010 $a3-319-30744-4 024 7 $a10.1007/978-3-319-30744-2 035 $a(CKB)3710000000717745 035 $a(DE-He213)978-3-319-30744-2 035 $a(MiAaPQ)EBC6314238 035 $a(MiAaPQ)EBC5594614 035 $a(Au-PeEL)EBL5594614 035 $a(OCoLC)953142637 035 $z(PPN)258870826 035 $a(PPN)19407918X 035 $a(EXLCZ)993710000000717745 100 $a20160505d2016 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aReal Analysis /$fby Peter A. Loeb 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2016. 215 $a1 online resource (XII, 274 p.) 311 $a3-319-30742-8 327 $aPreface -- Set Theory and Numbers -- Measure on the Real Line -- Measurable Functions -- Integration -- Differentiation and Integration -- General Measure Spaces -- Introduction to Metric and Normed Spaces -- Hilbert Spaces -- Topological Spaces -- Measure Construction -- Banach Spaces -- Appendices -- References. . 330 $aThis textbook is designed for a year-long course in real analysis taken by beginning graduate and advanced undergraduate students in mathematics and other areas such as statistics, engineering, and economics. Written by one of the leading scholars in the field, it elegantly explores the core concepts in real analysis and introduces new, accessible methods for both students and instructors. The first half of the book develops both Lebesgue measure and, with essentially no additional work for the student, general Borel measures for the real line. Notation indicates when a result holds only for Lebesgue measure. Differentiation and absolute continuity are presented using a local maximal function, resulting in an exposition that is both simpler and more general than the traditional approach. The second half deals with general measures and functional analysis, including Hilbert spaces, Fourier series, and the Riesz representation theorem for positive linear functionals on continuous functions with compact support. To correctly discuss weak limits of measures, one needs the notion of a topological space rather than just a metric space, so general topology is introduced in terms of a base of neighborhoods at a point. The development of results then proceeds in parallel with results for metric spaces, where the base is generated by balls centered at a point. The text concludes with appendices on covering theorems for higher dimensions and a short introduction to nonstandard analysis including important applications to probability theory and mathematical economics. . 606 $aFunctions of real variables 606 $aFunctional analysis 606 $aMeasure theory 606 $aReal Functions$3https://scigraph.springernature.com/ontologies/product-market-codes/M12171 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 606 $aMeasure and Integration$3https://scigraph.springernature.com/ontologies/product-market-codes/M12120 615 0$aFunctions of real variables. 615 0$aFunctional analysis. 615 0$aMeasure theory. 615 14$aReal Functions. 615 24$aFunctional Analysis. 615 24$aMeasure and Integration. 676 $a515 700 $aLoeb$b Peter A$4aut$4http://id.loc.gov/vocabulary/relators/aut$059462 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254074203321 996 $aReal analysis$91523597 997 $aUNINA