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.

Course II will plan to  include reviewing the knowledge of Lebesgue Integation and Integral Theorems (Convergence theorem (MCT、LDCT、BCT、UCT、 Fatou lemma、Tchebyshe inequality)，studying the Differentiation and Integration、L^P spaces and related Topics, Abstract Integration (Signed measure、Additive set measure、Radon-Nikodym theorem)，etc.

Details of the course: Below is the tentative learning schedule for this course, and the course content may be adjusted based on students' actual learning progress and outcomes.

1. Review the knowledge of Lebesgue Integration and Integration Theory {Convergence theorem (MCT、LDCT、BCT、UCT、 Fatou lemma、Tchebyshe inequality)。
2. The Repeated Integration: Fubini ， Tonelli theorem，Convolution. 
3. 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.
4. 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.
5. 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.
6. If the studying time is enough, we will furtherly study some topics of Abstract Integration (Signed measure、Additive set measure、Radon-Nikodym theorem, etc.