必修課程介紹

• 預備知識：advanced calculus
• 適合年級： 研究生
• 課程簡介：

本課程延續及推廣Riemann積分等微積分相關課程的理論並更進一步了解近代各分析學相關領域的發展，本課程將介紹測度論〈實數線上集合的長度或平面上集合的面積等理論的推廣〉、Lebesgue的積分理論〈Riemann的積分理論的推廣〉以及泛函分析的一些基本理論，作為分析學研究及其它相關應用領域之基礎

• 課程大綱：

1. Lebesgue measure and integral: Measurable functions, Littlewood’s three principles, Fatou’s lemma, monotone convergence theorem, bounded (dominated) convergence theorem.
2. Differentiation and integration: Function of bounded variation, absolute continuous function, singular function, fundamental theorem of calculus, Jensen ineguality.
3. Banach space：Lp spaces, Minkowski and Hölder inequalities, Riesz representation theorem.
4. (Optional) Topology: Baire category theory, Ascoli-Arzelá theorem, Stone-Weierstrass theorem.
5. General measure theory: Signed measure, Hahn decomposition theorem, Jordan decomposition Lebesgue Decomposition theorem, Radon-Nikodym theorem, Fubini theorem, Tonelli theorem.

• 參考書目：

1. H. L. Royden, Real Analysis, 3rd ed., 1989.
2. A. Friedman, Foundations of Modern Analysis.
3. R. L. Wheeden and A. Zygmund Measure and Integral.