MATH 540-Real Analysis
Time: MWF 2:00-2:50pm
Location: 140 Burrill Hall (Mathew Avenue)
Junge Office hours: Thursday 5-6 , Grader
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.
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
I: Review: Continuous functions and compact sets
Key words: basic definitions for metric spaces, continuous functions, and extension theorems.
II: Measure Theory
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
words: Simple functions, integration of positive functions,
space of integrable functions, Fatou's Lemma and the dominated
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
Video lectures :
Peter Loeb: Real Analysis, Birkhauser, first addition, 2016
H. L. Royden, Patrick Fitzpatrick: Real Analysis
Prentice Hall, 2010, fourth edition.
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)
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
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