실해석학

강의상세

수직선 상의 함수와 수열, Lebesque 측도와 적분을 다룬다. 수열의 극한, Cantor-like Sets를 다루고 Lebesque 측도의 정의와 성질, 측도 가능한 함수와 Lebesque 적분을 다룬다. 끝으로 함수의 미분 가능성도 다루게 된다.

1.	Review of chap Ⅰ and Ⅱ. The real number system.	1. basic properties of real numbers
2.	Outer measure and measurable sets	1. several propositions concerning algebras of sets.
3.	Lebesque measure and measurable sets	1. countably additive measure. 2. counting measure. 3. outer measure
4.	Nomeasurable sets	1. countably additive measure. 2. Lebesque measure
5.	Measurable functions	1. Borel sets and their measurability 2. Nomeasurable sets
6.	Littlewood's three principles	1. definition of measurable function 2. basic properties of Lebesgue measurable functions
7.	The Riemann Integral	1. some properties of Lebesgue measurable functions 2. Egoroff's theorem
8.	중간고사
9.	The Lebesque Integral of a bounded function	1. definition of Riemann integral 2 definition of Lebesgue integral
10.	The integral of a nonnegative function	1. Bounded convergence theorem 2. The integral of a nonnegative measurable function
11.	The general Lebesque Integral	1. Fatous lemma 2. monotone convergence theorem
12.	Convergences in measure	1. Lebesgue dominated convergence theorem 2. General L.D.C.T.
13.	The classical Banach spaces	1. normed linear spaces 2. definition of Lp-space 3. Banach spaces
14.	The Holder and Minkowski inequalities	1. The Holder inequalities 2. The Minkowski inequalities
15.	Bounded linear functionals	1. Rietz-Fischer theorem 2. Riesz representation theorem