MATH 540-Real Analysis

**Time:** MWF 2:00-2:50pm

**Location: ** 140 Burrill Hall (Mathew Avenue)

Course description :

Real Analysis, Math 540, is not only a comp course. It is formemost an introduction to basic topics in measure and integration theory which are use in many areas in mathematics.

One original goal is a modern (meaning modern in 1920) aporach to calculculus which is powerful and rigourous. Solving this issue is improtant in applications to PDE, Fourier analysis and analytic aspects of geometry (Riemian manifolds). We will also pursue another goal of making the notion of condition expectation in probability rigourous. Last, but not least, I will try to

demonstrate that some aspects of Hilbert space theory may be quite useful in solving real analysis problems. So the plan is to cover some beautiful basic theory, the comp exam related examples, and show there there is life after MAth 540.

I: Review: Continuous functions and compact sets

Key words: basic definitions for metric spaces, continuous functions, and extension theorems.

II: Measure Theory

Key words: Rings, sigma-algebras, measures, Caratheordory's extension theorem, product measures, Kolmogorov's 0-1 law, application to probability, non-measurable sets

II: Measurable Functions

Key words: Borel measurable functions, independent random variables, again Kolmogor, and Borel-Cantelli.

III: Integration Theory

Key words: Simple functions, integration of positive functions, space of integrable functions, Fatou's Lemma and the dominated convergence theorem

Vector-valued L_1

Connection to Riemann integral

V: Hilbert spaces

Key words: Basic definitions, Cauchy Schwarz, best approximation, existence of orthonormal basis.

IV: Banach spaces and Lp spaces

Key words: Defintion, K-functional, absolute continuity, Radon-Nikodym theorem, duality, covering lemma, differentiablity

VI: Bounded variation

Key words: Riesz representation theorem, singular measures

Examples and Cantor sets,

VII: Additional topics

Baire categorry

Old exam

Video lectures :

https://uofi.box.com/s/dnv0duoyqbk7r87ssm938lu6br9r3zmh

Books:

Peter Loeb: Real Analysis, Birkhauser, first addition, 2016

back up:

H. L. Royden, Patrick Fitzpatrick: Real Analysis

Prentice Hall, 2010, fourth edition.

Note: For additional reading both books are great, I will try the execrcies from Loeb's book, but make a foto of the page (just in case)

Grading

Homework (60%)The grader will only grade a selection of the submitted homework, and we will take someting like 8/11 for your grade. So you have a chance to learn and make mistakes, but this is a time intensive endeavor

Final (40%)

Monday 7:00-10:00 p.m., Friday, Dec. 14

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