Math
8100
 Real Analysis I  Fall 2021
Graduate
Real Analysis I
Tuesday and
Thursdays 11:1012:25 in Boyd 410
Office
Hours: TBA, during Math 8105 (see below), and by
appointment
(Math 8105 meets with me at
1:502:40 on Wednesdays in Boyd 410)
Syllabus for
2021
Old
Course Webpages from 2014 and 2018
and 2019
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
(discussion only, no
proofs)
 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 (in Math 8105)
Homework 1
(Due Thursday September 2)
 Lebesgue
Measurable Sets and Functions
 Lebesgue outer measure (Sections
1.1 and 1.2 in Stein)
 Preliminaries (decomposition theorems
for open sets)
 Properties of Lebesgue outer measure
 Lebesgue measure (Section
1.3 in Stein)
 Measurable sets form a sigmaalgebra
 Closed sets are measurable (only using
that countable unions of measurable sets
are measurable)
 Different characterizations of
Lebesgue measurability (using closed
sets, Gdelta sets and Fsigma sets)
 Countable additivity and the briefest of
introductions to abstract measure spaces
 Continuity from above and below
 Translation
invariance
Homework 2
(Due Tuesday September
14)
 Lebesgue measurable functions (Section
1.4 in Stein)
 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 in Stein)
 Every measurable set is nearly a
"very nice set", such as a finite union of
cubes or an open set
 The Lebesgue
Density Theorem is arguably one of
the strongest realizations of this
principle [Statement
only, proof is nonexaminable]
 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 Thursday September
23)
 Integration of nonnegative measurable
functions (Section 2.2 in Folland)
 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 and Fatou's Lemma
 Integration of extended realvalued and
complexvalued measurable functions (Section
2.3 in Folland)
 Space of all complexvalued integrable
functions form a complex vector space on
which the Lebesgue integral is a complex
linear functional
 Dominated Convergence Theorem
 Discussion on how to establish the
continuity and differentiablity of
functions defined by integrals
Homework 4
(Due Tuesday October 5)
 Definition of L^1, Completeness, and
interchanging sums and integrals
(Section 2.3 in Folland continued, see
also Section 2.4 in
Folland and Section 2.2 in Stein)
 Examples of different modes of
convergence
 Every sequence in L^1 which converges
in norm contains a subsequence that
converge almost everywhere
 Every absolutely convergent series in
L^1 converges almost everywhere (and in
L^1) to an L^1 function
 Conclusion of proof that L^1 is a
complete normed space (a Banach space)
 Discussion that more generally a
normed vector space X is complete if and
only every absolutely convergent series
in X converges
 Further Properties of the Lebesgue
Integral
 Relationship with Riemann integration
(Theorem 2.28 in Folland)
 Absolute continuity of the Lebesgue
integral and "small tails property"
(using the MCT as in proof of
Proposition 1.12 in Section 2.1 of
Stein)
 Translation and dilation invariance of
the Lebesgue integral (see
page 73 of Stein)
 An Approximation Theorem (Section
2.2 in Stein, see also Section
2.3 and 2.6 in Folland)
 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 (Proposition
2.5 in Section 2.2 of Stein)
 Short proofs of both the absolute
continuity and "small tails property" of
the Lebesgue integral
Exam 1
(In class on Thursday the 14th of October)
Old Exams for practice: 2019
(75 minutes), 2018
(60 minutes), 2014
(60 minutes), and 2013
(75 minutes)
 FubiniTonelli Theorem(s) (Section
2.3)
Homework 5 (Due Tuesday
October 26)
Homework 6 (Due Thursday
November 4)
Homework 7 (Due Monday
the 22nd of November)
 Basic Theory of L^p Spaces
 Function Theory of L^p Spaces
 Linear
Functionals and Dual spaces (for
general normed vector spaces)
 The
Dual Space of L^p (including a
sketch proof of the Riesz Representation
Theorem for L^p functions) ** We gave a different examinable
approach to the RRT for finite
measure spaces when 1=<p<2.
 ** Discussion on
the dual of L^∞ and the HahnBanach
theorem [did not happen so
nonexaminable]
 The
Riesz Representation Theorem (I will try
to discuss this after Thanksgiving
briefly)
Homework 8
(Due Thursday the 2nd of
December)
* We will not
have an inclass Exam 2. To help you study
for the final, here is Exam
2 from 2018, Exam
2 from 2014 and a Practice
Exam 2 (from 2014).
 Introduction
 Abstract measure spaces (definitions and
examples)
 Integration on measure spaces
 Absolute continuity and the
RadonNikodym theorem (vonNeumann's proof
using the RRT for L^2 spaces) [nonexaminable]
 ** Complex
measures (from Chapter 6 of
Rudin's Real and Complex Analysis) [not covered in
class, so nonexaminable]
 Construction of Measures:
 Outer measures, metric outer measures
and Caratheodory's theorem [nonexaminable]
 Regularity of
finite Borel measures
 (Caratheodory's)
Extension theorem
 Product measures
and Fubini/Tonelli
 Two Examples:
 Finite Borel measures and increasing
rightcontinous functions (the
LebesgueStieltjes integral) [nonexaminable]
 Application to
realizing the dual of C([0,1])
 Hausdorff
measure and dimension
 The Lebesgue
differentiation theorem [nonexaminable]
 Motivation from the
Fundamental Theorem of Calculus (Part
I) and the RadonNikodym Theorem [nonexaminable]
 The
HardyLittlewood maximal function [not covered in class]
 Proof of the
Lebesgue differentiation theorem [not covered in class]
 Pointwise
convergence of approximate
identities [not covered in class]
 Rademacher
functions, Random Series and the Strong
Law of Large Numbers
 Returning to Lebesgue's Theorem on the
Differentiation of a Monotone Function [nonexaminable]
 Monotone rightcontinuous functions are
differentiable almost everywhere (using
the associated LebesgueStieltjes measure,
RadonNikodym and Lebesgue differentiation
theorems) [only
briefly discussed with no proofs,
nonexaminable]
 Discussion of the notion of absolute
continuity [nonexaminable]
 The Fundamental Theorem of Calculus
(Part II) [nonexaminable]
