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)

  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.
We may cover the following.
  1. Lp spaces: Minkowski's inequality and Holder's inequality, completeness of Lp, denseness results in Lp.
  2. Fourier Series: definition of Fourier Series, formulation of convergence problems, the L2 theory of Fourier series, convergence of Fourier Series.


  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