I.Contents:
Part1:Introduction
Summability of real numbers. A quick introduction to measure and integration.
Part2: Outer measures
Measures arising from outer measures, regularity of outer measures, 1-dimensional Lebesgue measure, Caratheodory outer measures, n-dimensional Lebesgue measure and transformation of integrals.
Part3: L昌-spaces
Completeness of L昌-spaces, Orthonormal systems-completeness and Fourier expansion. Conditional expectation.
Part4: Decomposition and Differentiation of Measures
Signed measures, The Lebesgue-Radon-Nikodym theorem, Conditional expectation, Differentiation on R明, Functions of bounded variation, Convex functions.
II.Course prerequisite:
Advanced Calculus, Linear Algebra.
III.Reference material ( textbook(s) ):
B. R. Gelbaum, Modern Real and Complex Analysis.
E. H. Lieb & M. Loss, Analysis.
S. Saks, Theory of the Integral.
R. L. Wheeden & A. Zygmund, Measure and Integral.
IV.Grading scheme:
Home work and Mid-term examination each counts 30%
Final examination counts 40%
V.Others: