MATH 320 Homepage (Fall 2018)


Lecturer: Alexandru Ionescu, aionescu@math.princeton.edu


Handouts

Information.
Problem set 1. Due in class on Thursday, September 20.
Problem set 2. Due in class on Thursday, September 27.
Problem set 3. Due in class on Thursday, October 4.
Problem set 4. Due in class on Thursday, October 11.
Problem set 5. Due in class on Thursday, October 18.
Practice Midterm Exam. Not to be turned in.
Problem set 6. Due in class on Tuesday, November 13.
Problem set 7. Due in class on Tuesday, November 20.
Problem set 8. Due in class on Thursday, November 29.
Problem set 9. Due in class on Thursday, December 6.
Problem set 10. Due in class on Thursday, December 13.

Syllabus

Sep. 13: Introduction, real numbers.
Sep. 15: The natural numbers, countable and uncountable sets.
Sep. 20: Open sets and closed sets on the real line, compactness.
Sep. 25: The nested set theorem, sequences of real numbers, the Bolzano-Weierstrass Theorem.
Sep. 27: Upper and lower limits of sequences, continuous functions.
Oct. 02: The Intermediate Value Theorem, uniformly continuous functions, outer Lebesgue measure.
Oct. 04: Outer Lebesgue measure, Lebesgue measurable sets.
Oct. 09: The \sigma-algebra of Lebesgue measurable sets, outer and inner approximations.
Oct. 11: Continuity of measure, nonmeasurable sets, Cantor sets.
Oct. 16: Lebesgue measurable functions, basic properties, the Cantor-Lebesgue function.
Oct. 18: Pointwise limits of measurable functions, approximation by simple functions, Egoroff's theorem.
Oct. 23: Lusin's theorem, preliminaries of Lebesgue integration.
Oct. 25: MIDTERM EXAM.
Nov. 06: Lebesgue integration of bounded functions over sets of finite measure.
Nov. 08: Linearity, the bounded convergence theorem, Fatou's lemma, the monotone convergence theorem.
Nov. 13: Integrable functions, the Lebesgue Dominated Convergence Theorem.
Nov. 15: Vitali convergence theorems, convergence in measure.
Nov. 20: Convergence in measure vs. pointwise convergence, continuity of monotone functions.
Nov. 27: Differentiability of increasing functions, Lebesgue's theorem.
Nov. 29: The Vitali covering, functions of bounded variation.
Dec. 04: Absolutely continuous functions, the fundamental theorem of calculus.
Dec. 06: Antiderivatives of integrable functions, convex functions.
Dec. 11: Normed spaces, the L^p spaces, the inequalities of Holder and Minkowski.