FALL 2026
Mariusz Mirek
Course Description:
While the term “real analysis” initially referred to the theory of functions of a real variable, it has evolved to encompass a broader range of more general and abstract topics that form the foundation of modern analysis. This course will explore these fundamental theories and their applications.
We will cover a variety of topics, including:
1) Sigma-algebras and measures
2) Outer measures and Borel measures on the real line
3) Lebesgue measure on the real line and measurable functions
4) Integration of non-negative and complex functions
5) Modes of convergence
6) Finite product measures and the d-dimensional Lebesgue measure and integral
7) Integration in polar coordinates
8) Infinite product measures and signed measures
9) The Lebesgue–Radon–Nikodym theorem
10) Complex measures and differentiation in Euclidean space
11) Functions of bounded variation
12) Hilbert spaces and Lp spaces
Through this comprehensive curriculum, students will develop a deep understanding of both the theoretical underpinnings and practical applications of real analysis, essential for further study in mathematics and related fields.
REVIEW SESSIONS: TBD.
Text:
“Real Analysis: Modern Techniques and Their Applications” by Gerald B. Folland
Prerequisite:
A solid understanding of the classical theory of functions of a real variable, including limits and continuity, differentiation and Riemann integration, infinite series, and uniform convergence is essential. Additionally, familiarity with the arithmetic of complex numbers and some elementary set theory and topology is required. A basic knowledge of linear algebra, focusing on the definitions of vector spaces, linear mappings, and determinants, is also expected. This foundational knowledge is crucial for success in the course.
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FALL 2025
Ian Jauslin
Course Description:
This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
REVIEW SESSIONS: TBD.
Text:
``Real Analysis'' by Folland
Prerequisite:
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FALL 2024
Eric Carlen
Course Description:
This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
REVIEW SESSIONS: TBD.
Text:
``Real Analysis'' by Folland
Prerequisite:
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FALL 2023
Mariusz Mirek
Course Description:
This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
REVIEW SESSIONS: TBD.
Text:
``Real Analysis'' by Folland
Prerequisite:
Good knowledge of linear algebra and some acquaintance with abstract algebra (say first semester of Abstract Algebra for graduate students)
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FALL 2022
Michael Kiessling
Course Description:
This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
REVIEW SESSIONS: TBD
Text:
``Real Analysis'' by Folland
Prerequisite:
The classical theory of functions of a real variable: point-set topology, limits and continuity, differentiation and Riemann integration, infinite series, uniform convergence
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FALL 2021
Li-Cheng Tsai
Course Description:
This is a course aiming at graduate students in mathematics. This course will focus on measure theory and integration, covering the topics of Lebesgue measure and integral, abstract measure and integral, L2 and Lp spaces, elementary Hilbert space theory, absolute continuity, Radon-Nikodym theorem, product measure and integral.
REVIEW SESSIONS: TBD
Text:
Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein and Rami Shakarchi
Prerequisite:
The classical theory of functions of a real variable: point-set topology, limits and continuity, differentiation and Riemann integration, infinite series, uniform convergence
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Schedule of Sections:
16:640:501 Schedule of Sections