Math 241: Real Analysis
Instructor: J. T. Beale

This course develops measure theory and Lebesgue integration. This theory is
foundational for much of analysis and in particular for the mathematical
theory of probability.  This course is a natural beginning part of a
graduate program in mathematics.  Besides measure theory and integration, we
include a brief treatment of Fourier series and transforms, and the
formulation of probability theory in terms of measure theory, including
proofs of basic theorems such as the Central Limit Theorem.

The course is aimed at beginning graduates students in mathematics, but
other interested students are welcome if they have the necessary background.
The main prerequisite is a rigorous undergraduate course in real analysis.,
such as our Math 203 (preferable) or our Math 139.   Students should have a
working knowledge of the concepts from undergraduate analysis and should be
used to writing proofs in analysis.  Specifically, students should be
familiar with these concepts: algebra of sets and images and inverse images
of functions, sup and inf of sets of real numbers, countable and uncountable
sets, completeness of the real numbers, Cauchy sequences, metric spaces,
open and closed sets, compact sets (as defined by coverings by open sets),
the Bolzano-Weierstrass Theorem, uniform continuity as distinguished from
continuity, the relation between continuous functions and compact or
connected sets, pointwise convergence and uniform convergence of sequences
of functions, convergence of power series, the Riemann integral as defined
as the limit of upper and lower sums, and the proofs of standard theorems of
calculus.

Please note!!  The textbook will be the NEW fourth edition of the book Real
Analysis by Royden and Fitzpatrick.  The earlier editions were by Royden
alone. The new edition is significantly different.  Unfortunately this
important distinction is not evident in the bookstore listing.