Math 404 A: Analysis II

This directory contains the notes and the homework statements for Math 404A: Analysis II.

Textbook

List of Topics (from the Syllabus)

1. Lebesgue Measure on Rⁿ: Introduction, outer measure, measurable sets, Lebesgue measure, regularity properties, a nonmeasurable set. Measurable functions, Egorov's theorem, Lusin's theorem.
2. Lebesgue Integration: Simple functions, Lebesgue integration of a bounded function over a set of finite measure. Bounded Convergence Theorem, integral of nonnegative functions, Fatou's lemma, Monotone Convergence Theorem, Lebesgue's Dominated Convergence Theorem. Change of variables formula.
3. Differentiation and Integration: Functions of bounded variation, Differentiation of an integral, absolute continuity.

1. L^p spaces: Minkowski's inequality and Holder's inequality, completeness of L^p, denseness results in L^p.
2. Fourier Series: definition of Fourier Series, formulation of convergence problems, the L² theory of Fourier series, convergence of Fourier Series.

Notes

1. Preliminaries.   Homework 1
2. Lebesgue Measure on ℝ   Homework 2
3. Measurable Functions   Homework 3
4. Lebesgue Integration   Homework 4
5. Fubini's Theorem
6. Differentiation and Lebesgue Integration   Homework 5

References

• Thomas Hawkins, "Lebesgue's Theory of Integration: Its Origins and Development", AMS Chelsea Publications, Providence, Rhode Island, USA, 2000.
• Gerald A. Edgar, "Classics on Fractals", Westview Press, 2003.