LEADER 03985nam 22005175 450 001 9911031671803321 005 20251001130603.0 010 $a981-9527-58-9 024 7 $a10.1007/978-981-95-2758-8 035 $a(CKB)41521165200041 035 $a(MiAaPQ)EBC32324975 035 $a(Au-PeEL)EBL32324975 035 $a(DE-He213)978-981-95-2758-8 035 $a(EXLCZ)9941521165200041 100 $a20251001d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aElements of Measure and Probability /$fby Arup Bose 205 $a1st ed. 2025. 210 1$aSingapore :$cSpringer Nature Singapore :$cImprint: Springer,$d2025. 215 $a1 online resource (318 pages) 225 1 $aTexts and Readings in Mathematics,$x2366-8725 ;$v88 311 08$a981-9527-57-0 327 $aPreliminaries -- Classes of Sets -- Introduction to Measures -- Extension of Measures -- Lebesgue-Stieltjes Measures -- Measurable Functions -- Integral -- Basic Inequalities -- Lp Spaces: Topological Properties -- Product Spaces and Transition Measures -- Random Variables and Vectors -- Moments and Cumulants -- Further Modes of Convergence of Functions -- Independence and Basic Conditional Probability -- 0-1 Laws -- Sums of Independent Random Variables -- Convergence of Finite Measures -- Characteristic Functions -- Central Limit Theorem -- Signed Measure -- Randon-Nikodym Theorem -- Fundamental Theorem of Calculus -- Conditional Expectation. 330 $aThis book can serve as a first course on measure theory and measure theoretic probability for upper undergraduate and graduate students of mathematics, statistics and probability. Starting from the basics, the measure theory part covers Caratheodory?s theorem, Lebesgue?Stieltjes measures, integration theory, Fatou?s lemma, dominated convergence theorem, basics of Lp spaces, transition and product measures, Fubini?s theorem, construction of the Lebesgue measure in Rd, convergence of finite measures, Jordan?Hahn decomposition of signed measures, Radon?Nikodym theorem and the fundamental theorem of calculus. The material on probability covers standard topics such as Borel?Cantelli lemmas, behaviour of sums of independent random variables, 0-1 laws, weak convergence of probability distributions, in particular via moments and cumulants, and the central limit theorem (via characteristic function, and also via cumulants), and ends with conditional expectation as a natural application of the Radon?Nikodym theorem. A unique feature is the discussion of the relation between moments and cumulants, leading to Isserlis? formula for moments of products of Gaussian variables and a proof of the central limit theorem avoiding the use of characteristic functions. For clarity, the material is divided into 23 (mostly) short chapters. At the appearance of any new concept, adequate exercises are provided to strengthen it. Additional exercises are provided at the end of almost every chapter. A few results have been stated due to their importance, but their proofs do not belong to a first course. A reasonable familiarity with real analysis is needed, especially for the measure theory part. Having a background in basic probability would be helpful, but we do not assume a prior exposure to probability. 410 0$aTexts and Readings in Mathematics,$x2366-8725 ;$v88 606 $aMeasure theory 606 $aProbabilities 606 $aMeasure and Integration 606 $aProbability Theory 615 0$aMeasure theory. 615 0$aProbabilities. 615 14$aMeasure and Integration. 615 24$aProbability Theory. 676 $a515.42 700 $aBose$b Arup$0768007 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911031671803321 996 $aElements of Measure and Probability$94443102 997 $aUNINA