MA442 Real Variables


Instructor: Nikolay S. Strigul
E-mails: nstrigul@stevens.edu
Office Hours: by appointment

Lectures: TR, 12:00-01:15 pm, Babbio Center 221
Homeworks: There will be weekly homework assignments.
Exams: There will be Midterm and Final Exams.
Quizzes: There will be regular quizzes covering definitions and theorems.

Grading:

Course program Course program (PDF)

General comments:

MA442 is an advanced undegraduate course in real analysis. The major objective of this course is to cover Lebesgue measure and integral. Our major compromise in order to deliver this theory to undegraduates is to construct Lebesgue's theory on the real line rather than in an abstract space (we also will introduce it on the real plane). For the same reason, necessary background topics from algebra and topology will be introduced in the context of metric and normed vector spaces. Lp spaces will be also considered in this course. The theory will be generalized if the time permits.

Textbooks:

1) Introduction to Real Analysis by A.N. Kolmogorov and S.V. Fomin
2) Real Analysis by N. L. Carothers
3) Theory of Functions of a Real Variable by I.P. Natanson
4) A Short Course of the Theory of Functions of a Real Variable by B.Z. Vulih (available only in Russian)

Course program:

Lecture 1. - Groups, rings and fields. Real numbers.

Lecture 2. - Set theory 1. Axiomatic. Equivalence relations.

Lecture 3. - Set theory 2. Order relations.

Lecture 4. - Set theory 3. Cardinality.

Lecture 5. - Metric spaces.

Lecture 6. - Semirings, rings and algebras of sets.

Lecture 7. - Normed vector spaces.

Lecture 8. - Topology. Open and closed sets.

Lecture 9. - Continuous functions.

Lecture 10. - Homeomorphisms.

Lecture 11. - Connectedness.

Lecture 12. - Completeness.

Lecture 13. - Compactness.

Lecture 15. - Sequences of functions.

Lecture 16. - Lebesgue measure 1.

Lecture 17. - Lebesgue measure 2.

Lecture 18. - Lebesgue measure 3.

Lecture 19. - Measurable functions 1.

Lecture 20. - Measurable functions 2.

Lecture 21. - Measurable functions 3.

Lecture 22. - The Lebesgue integral 1.

Lecture 23. - The Lebesgue integral 2.

Lecture 24. - The Lebesgue integral 3.

Lecture 25. - Convergence in measure.

Lecture 26. - The L2 spaces.

Lecture 27. - The Lp spaces.

Lecture 28. - Sequences of functions. Lebesgue's differentiation theorem.

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