\contentsline {section}{\numberline {1}Lecture 1 -- Measures and sigma-Fields}{4}{section.1}%
\contentsline {subsection}{\numberline {1.1}Notation and set-theoretic preliminaries}{4}{subsection.1.1}%
\contentsline {subsection}{\numberline {1.2}$\sigma $-fields}{4}{subsection.1.2}%
\contentsline {subsection}{\numberline {1.3}Measures}{5}{subsection.1.3}%
\contentsline {subsection}{\numberline {1.4}Special classes of measures}{5}{subsection.1.4}%
\contentsline {subsection}{\numberline {1.5}Examples on a finite sample space}{6}{subsection.1.5}%
\contentsline {section}{\numberline {2}Lecture 2 -- Constructing sigma-Fields and Measures}{7}{section.2}%
\contentsline {subsection}{\numberline {2.1}Semirings, rings, and fields}{7}{subsection.2.1}%
\contentsline {subsection}{\numberline {2.2}Set functions and pre-measures}{8}{subsection.2.2}%
\contentsline {subsection}{\numberline {2.3}Outer measure and $\mu ^{*}$-measurability}{8}{subsection.2.3}%
\contentsline {subsection}{\numberline {2.4}Carath\'eodory's extension theorem}{9}{subsection.2.4}%
\contentsline {section}{\numberline {3}Lecture 3 -- Uniqueness; Dynkin's $\pi $--$\lambda $; Completeness}{11}{section.3}%
\contentsline {subsection}{\numberline {3.1}$\pi $- and $\lambda $-systems}{11}{subsection.3.1}%
\contentsline {subsection}{\numberline {3.2}Dynkin's $\pi $--$\lambda $ theorem}{11}{subsection.3.2}%
\contentsline {subsection}{\numberline {3.3}Uniqueness of extension}{12}{subsection.3.3}%
\contentsline {subsection}{\numberline {3.4}Completeness}{12}{subsection.3.4}%
\contentsline {section}{\numberline {4}Lecture 4 -- Lebesgue Measure; Non-Measurable Sets}{14}{section.4}%
\contentsline {subsection}{\numberline {4.1}Lebesgue measure on $(0,1]$ and $\vvmathbb {R}$}{14}{subsection.4.1}%
\contentsline {subsection}{\numberline {4.2}Non-measurable sets: the Vitali construction}{15}{subsection.4.2}%
\contentsline {subsection}{\numberline {4.3}Product measures, briefly}{16}{subsection.4.3}%
\contentsline {subsection}{\numberline {4.4}Independence}{16}{subsection.4.4}%
\contentsline {section}{\numberline {5}Lecture 5 -- Simple and Measurable Functions}{18}{section.5}%
\contentsline {subsection}{\numberline {5.1}Simple random variables}{18}{subsection.5.1}%
\contentsline {subsection}{\numberline {5.2}Simple functions on a measure space}{19}{subsection.5.2}%
\contentsline {subsection}{\numberline {5.3}Measurable functions}{20}{subsection.5.3}%
\contentsline {subsection}{\numberline {5.4}Useful facts about measurability}{20}{subsection.5.4}%
\contentsline {subsection}{\numberline {5.5}Almost everywhere}{21}{subsection.5.5}%
\contentsline {section}{\numberline {6}Lecture 6 -- Integration and Convergence Theorems}{23}{section.6}%
\contentsline {subsection}{\numberline {6.1}Lebesgue integration for measurable functions}{23}{subsection.6.1}%
\contentsline {subsection}{\numberline {6.2}The three convergence theorems}{24}{subsection.6.2}%
\contentsline {section}{\numberline {7}Lecture 7 -- Lebesgue-Stieltjes Measure; Fubini-Tonelli}{26}{section.7}%
\contentsline {subsection}{\numberline {7.1}Image measures and Lebesgue--Stieltjes}{26}{subsection.7.1}%
\contentsline {subsection}{\numberline {7.2}Product $\sigma $-fields and the product measure}{27}{subsection.7.2}%
\contentsline {subsection}{\numberline {7.3}The monotone class theorem}{28}{subsection.7.3}%
\contentsline {subsection}{\numberline {7.4}The Fubini--Tonelli theorem}{28}{subsection.7.4}%
\contentsline {section}{\numberline {8}Lecture 8 -- Lp Spaces and Classical Inequalities}{30}{section.8}%
\contentsline {subsection}{\numberline {8.1}The spaces $L^p$}{30}{subsection.8.1}%
\contentsline {subsection}{\numberline {8.2}Markov, Chebyshev, Chernoff}{30}{subsection.8.2}%
\contentsline {subsection}{\numberline {8.3}Convexity and Jensen's inequality}{31}{subsection.8.3}%
\contentsline {subsection}{\numberline {8.4}H\"older and Minkowski}{32}{subsection.8.4}%
\contentsline {subsection}{\numberline {8.5}Approximation in $L^p$}{32}{subsection.8.5}%
\contentsline {section}{\numberline {9}Lecture 9 -- Convergence in Probability and Measure}{34}{section.9}%
\contentsline {subsection}{\numberline {9.1}Weak convergence of probability measures}{34}{subsection.9.1}%
\contentsline {subsection}{\numberline {9.2}Random variables and their distributions}{35}{subsection.9.2}%
\contentsline {subsection}{\numberline {9.3}Modes of convergence}{35}{subsection.9.3}%
\contentsline {subsection}{\numberline {9.4}Hierarchy of convergence}{36}{subsection.9.4}%
\contentsline {section}{\numberline {10}Lecture 10 -- Hierarchy of Convergence; Borel-Cantelli}{38}{section.10}%
\contentsline {subsection}{\numberline {10.1}Stronger metrics on the space of probability measures}{38}{subsection.10.1}%
\contentsline {subsection}{\numberline {10.2}Hierarchy of modes of convergence}{38}{subsection.10.2}%
\contentsline {subsection}{\numberline {10.3}Limsup, liminf, and the Borel--Cantelli setup}{39}{subsection.10.3}%
\contentsline {subsection}{\numberline {10.4}The Borel--Cantelli lemmas}{39}{subsection.10.4}%
\contentsline {subsection}{\numberline {10.5}Prohorov's theorem}{40}{subsection.10.5}%
\contentsline {section}{\numberline {11}Lecture 11 -- Law of Large Numbers}{41}{section.11}%
\contentsline {subsection}{\numberline {11.1}Setup: independence and identical distribution}{41}{subsection.11.1}%
\contentsline {subsection}{\numberline {11.2}Weak law of large numbers}{41}{subsection.11.2}%
\contentsline {subsection}{\numberline {11.3}Strong law of large numbers}{42}{subsection.11.3}%
\contentsline {section}{\numberline {12}Lecture 12 -- Central Limit Theorem; Characteristic Functions}{43}{section.12}%
\contentsline {subsection}{\numberline {12.1}Gaussian measures}{43}{subsection.12.1}%
\contentsline {subsection}{\numberline {12.2}Characteristic functions}{43}{subsection.12.2}%
\contentsline {subsection}{\numberline {12.3}L\'evy's continuity lemma}{44}{subsection.12.3}%
\contentsline {subsection}{\numberline {12.4}The central limit theorem}{44}{subsection.12.4}%
\contentsline {section}{\numberline {13}Lecture 13 -- The Ergodic Theorem}{47}{section.13}%
\contentsline {subsection}{\numberline {13.1}Measure-preserving maps, invariance, ergodicity}{47}{subsection.13.1}%
\contentsline {subsection}{\numberline {13.2}Ergodic theorems}{48}{subsection.13.2}%
\contentsline {subsection}{\numberline {13.3}Application: the strong law of large numbers, again}{49}{subsection.13.3}%