Math
8100
 Real Analysis I  Fall 2019
Graduate
Real Analysis I
Tuesday and
Thursdays 11:0012:15 in Boyd 303
Office
Hours: MF 1:252:15, during Math 8105 (see
below), and by appointment
(Math 8105 meets 1:252:15
in Boyd 410)
Syllabus for
2019
Old
Course Webpages from 2014 and 2018
Principal
Textbook:
Real
Analysis, by E. M. Stein and R.
Shakarchi
Secondary
References:
Real Analysis,
by G. B. Folland
Real and Complex
Analysis, by W. Rudin
Course Outline (eventually including Assignments) and Some Supplementary Class Notes
For the most
part we shall follow closely the appropriate
sections of the reference(s) listed above.
 Three Notions of Smallness for subsets of
the Reals
 Countable, Meager (1st Category), and
Null (measure zero)
 The Reals are "not small", specifically
they are neither countable, meager, or
null (Theorems of Cantor, Baire, and Borel
respectively)
 Discontinuities and NonDifferentiability
 Fsigma sets, first class functions,
and Lebesgue's Criterion for Riemann
Integrability
 Nowhere differentiable functions and
Lebesgue's Theorem on the differentiablity
of monotone functions
 Review of Uniform Convergence and Uniform
Continuity
Homework 1
(Due Tuesday August 27)
 Lebesgue
Measurable Sets and Functions
 Lebesgue outer measure (Sections
1.1 and 1.2)
 Preliminaries (decomposition theorems
for open sets)
 Properties of Lebesgue outer measure
 Lebesgue measure (Section
1.3)
Homework 2
(Due Thursday September
5)
 Lebesgue measurable functions (Section
1.4)
 Equivalent definitions and the useful
characterization using open sets
 Compositions and closure under algebraic
and limiting operations
 Approximation by simple functions (and
step functions)
 Littlewood's Three Principles (Section
1.4)
 Every measurable set is nearly an
open set (the Lebesgue Density Theorem is
arguably the strongest realization of this
principle)
 Every measurable function is nearly a
continuous function (Lusin's theorem)
 Every convergent sequence of measurable
functions is nearly uniformly
convergent (Egorov's theorem)
Homework 3
(Due Friday September
20)
 Integration of nonnegative measurable
functions (Section 2.1)
 Simple functions and their integral
(including properties such as monotonicity
and linearity)
 Extension to all nonnegative measurable
functions (establishing linearity using
MCT below)
 Monotone Convergence Theorem (proved
using "continuity from below" for
measures defined by simple functions)
 Chebyshev's inequality and proof that
"f equals 0 a.e. if and only if
the integral of f equals 0"
 Convergence Theorems for nonnegative
measurable functions and examples
 (Modified) Monotone Convergence
Theorem, Fatou's Lemma, and the
Dominated Convergence Theorem
 Discussion that these results are
"equivalent"
 Integration of extended realvalued and
complexvalued measurable functions
 Provisional definition of L^1 and "the"
Dominated Convergence Theorem
 Discussion on how to establish the
continuity and differentiablity of
functions defined by integrals
 Absolute continuity of the Lebesgue
integral and "small tails property" (using
the Monotone Convergence Theorem)
 Translation and dilation invariance of
the Lebesgue integral
 Relationship with Riemann integration
Homework 4 (Due
Tuesday October 1)
 Completeness of L^1 and interchanging sums
and integrals
 Every absolutely convergent series in
L^1 converges almost everywhere (and in
L^1) to an L^1 function
 Examples of different modes of
convergence
 Sequences in L^1 which converge in
norm contain subsequences that converge
almost everywhere
 Conclusion of proof that L^1 is a
complete normed space (a Banach space)
 An Approximation Theorem
 Simple functions and Continuous
functions with compact support are both
dense in L^1  another realization of
Littlewood's second principle
 Continuity in L^1
 Short proofs of both the absolute
continuity and "small tails property" of
the Lebesgue integral
