Math 404 A: Analysis II
This directory contains the notes and the homework statements for Math
404A: Analysis II.
Textbook
Real Analysis, Royden and Fitzpatrick.
List of Topics (from the Syllabus)
 Lebesgue Measure on R^{n}:
Introduction, outer measure, measurable sets, Lebesgue measure,
regularity properties, a nonmeasurable set.
Measurable functions, Egorov's theorem, Lusin's theorem.
 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.
 Differentiation and Integration:
Functions of bounded variation, Differentiation of an integral,
absolute continuity.
We may cover the following.
 L^{p} spaces:
Minkowski's inequality and Holder's inequality, completeness of
L^{p}, denseness results in L^{p}.
 Fourier Series: definition of Fourier Series, formulation
of convergence problems, the L^{2} theory of Fourier
series, convergence of Fourier Series.
Notes
 Preliminaries.
Homework 1
 Lebesgue
Measure on ℝ
Homework 2
 Measurable
Functions
Homework 3
 Lebesgue
Integration
Homework 4

Fubini's Theorem

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.