Course outline  This is an EMI course. This course I and course II during two semsters provide a rigorous treatment of fundamental topics in real analysis, including Measure Theory、Lebesgue Measurable Functions、Lebesgue Integation and Integral Theorems、Differentiation and Integration、L^P spaces and related Topics, Hilbert Space and Metric Spaces*, etc.

Details of the course In this course we will study the following topics:

1. Preliminaries: The Real Numbers: Sets, Sequences, and Functions
2. Measure Theory: Preliminaries, Outer-Measure, Lebesgue Measurable Sets, Properties of Measurable Sets, Countable Additivity and Continuity of Measure, and the Borel-Cantelli Lemma, Vitali's Example of a Non-measurable Set, The Cantor Set and the Cantor-Lebesgue Function . 
3. Lebesgue Measurable Functions: Sums, Products, and Compositions of Measurable Functions; Pointwise Limits, Simple Approximation; Littlewood's Three Principles, Egoro 's Theorem, Lusin's Theorem.
4. Lebesgue Integration and Integration Theory: Review of Riemann Integral; The Integral of a Bounded, Finitely Supported; The Integral of a Non-Negative Measurable Function and The General Lebesgue Integral; Countable Additivity and Continuity of Integration; Uniform Integrability and Tightness: The Vitali Convergence Theorems; Convergence in the Mean and in Measure (A Theorem of Riesz); Characterizations of Riemann and Lebesgue Integrability
5. Differentiation and Integration: Continuity and Differentiability of Monotone Function; Lebesgue's Theorem; Functions of Bounded Variation: Jordan's Theorem; Absolutely Continuous Functions; Integrating Derivatives: Differentiating Indefinite Integrals; Measurability of Images of Sets, Compositions of Functions; Convex Functions.
6. The L^p Spaces: Completeness and Approximation; Normed Linear Spaces; The Inequalities of Young, Holder, and Minkowski; 
   L^p Is Complete: Rapidly Cauchy Sequences and the Riesz-Fischer Theorem Approximation and Separability.
7. Further Topics of The L^p Spaces: Duality, Weak Convergence, and Minimization; Bounded Linear Functionals on a Normed Linear Space; The Riesz Representation Theorem for the Dual of L^p; Weak Sequential Convergence in L^p; The Minimization of Convex Functionals.
8. Hilbert Spaces*: An Introduction, The Hilbert space L2, Hilber spaces, Fourier series and Fatou‘s theorem, Closed subspaces and orthogonal projections, Linear functionals and Riesz representation theorem, Adjoints.

References: (1) H. L. Royden, Real Analysis.

                   (2) A. Zygmund, Measure Theory and Integrations, New Edition.