## Topics: - $\sigma$-algebra and measures: definition of measures, outer measures, Caratheodory theorem, extension of measures and construction of Lebesgue measure, Borel measures on $\mathbb{R}$ and $\mathbb{R}^n$. - Integration and $L^p$ space: measurable functions, definition of integrals, $L^1$ space, modes of convergence, product measure and Fubini-Tonelli Theorem. $L^p$ space and inequalities in $L^p$ spaces. - Signed measure, Lebesgue-Radon-Nykodim theorem, differentiation and absolute continuity, Hardy-LIttlewood maximal function. - Some topological prerequisites: topology, neighborhood, countability and separation axioms, continuity. - $L^p$ space as Banach spaces: introduction to Banach spaces, linear functionals, dual of $L^p$ spaces. - Continuous functions and $L^p$ functions: LCH spaces, Stone-Weierstrass theorem, Radon measures, the Riesz-Markov Theorem. - Supplementary materials (optional): Haar measure, Hausdorff measure and self similar measures. ## Textbook reference: Folland, Real Analysis: Modern Techniques and Their Applications. Wiley, 1999. We will cover the following sections in Folland (although the presentation may differ and we will not cover them in the same order): (1.1) - (1.5), (2.1) - (2.6), (3.1) - (3.4), part of (3.5), (5.1) - (5.2), (6.1) - (6.2), part of (7.1) - (7.3). Optional: (11.1) - (11.3). ## References - Royden, Fitzpatrck, Real Analysis. Prentice Hall, 2010. + A complete reference including set theoretic axioms and real number theory. + Contains more topics than we cover, good for supplementary reading. - Halmos, Measure Theory. Springer, 2013. + Detailed, slow paced exposition of measure theory. - Tao, An Introduction to Measure Theory. American Mathematical Society. 2011. + Starts with $\mathbb{R}^n$. + Many exercises. + A shorter book based on a quarter course. ## Evaluation * Homework: 30% * Midterm Test: 30% * Final Exam: 40%