Math 404 A: Analysis II
This directory contains the notes and the homework statements for Math
404A: Analysis II.
Real Analysis, Royden and Fitzpatrick.
List of Topics (from the Syllabus)
We may cover the following.
- Lebesgue Measure on Rn:
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
Change of variables formula.
- Differentiation and Integration:
Functions of bounded variation, Differentiation of an integral,
- Lp spaces:
Minkowski's inequality and Holder's inequality, completeness of
Lp, denseness results in Lp.
- Fourier Series: definition of Fourier Series, formulation
of convergence problems, the L2 theory of Fourier
series, convergence of Fourier Series.
Measure on ℝ
Differentiation and Lebesgue Integration
- Thomas Hawkins, "Lebesgue's Theory of Integration: Its Origins and
Development", AMS Chelsea Publications, Providence, Rhode Island,
- Gerald A. Edgar, "Classics on Fractals", Westview Press,