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 in Stein)
 Preliminaries (decomposition theorems
for open sets)
 Properties of Lebesgue outer measure
 Lebesgue measure (Section
1.3 in Stein)
Homework 2
(Due Thursday September
5)
 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) [Statement
only in case of step functions, proof is
nonexaminable for now]
 Littlewood's Three Principles (Section
1.4 in Stein)
 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) [Statement only, proof
is nonexaminable for now]
 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 2.2 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, Fatou's Lemma, and the
Dominated Convergence Theorem
 Discussion that these results are
"equivalent"
 Integration of extended realvalued and
complexvalued measurable functions (Section
2.3 in Folland)
 Provisional definition of L^1 and "the"
Dominated Convergence Theorem
 Discussion on how to establish the
continuity and differentiablity of
functions defined by integrals
 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)
Homework 4
(Due Tuesday October 1)
 Completeness of L^1 and interchanging sums
and integrals (Section 2.3 in
Folland continued, see also Section
2.4 in Folland and Section
2.2 in Stein)
 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
 Every sequence in L^1 which converges
in norm contains a subsequence that
converge almost everywhere
 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
 An Approximation Theorem (Section
2.2 in Stein, see also Section
2.3 and 2.6 in Folland)
[nonexaminable
for now]
 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 10th of October)
Here is Exam
1 from 2018 (60 mins), Exam
1 from 2014 (60 mins), and Exam
1 from 2013 (75 mins) for practice
 FubiniTonelli Theorem(s) (Section
2.3)
Homework 5 (Due Friday
October 18)
Homework 6 (Due Thursday
October 31)
Homework 7 (Due Thursday
the 14th 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
approach to the RRT for p=1
 ** Discussion on
the dual of L^∞ and the HahnBanach
theorem [nonexaminable]
 The
Riesz Representation Theorem (statements
only)
Homework 8
(Due Tuesday the 26th of
November)
Exam
2
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)
 ** Complex
measures (from Chapter 6 of
Rudin's Real and Complex Analysis) [not covered in
class]
 Construction of
Measures:
 Outer measures,
metric outer measures and Caratheodory's
theorem
 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)
 Application to
realizing the dual of C([0,1])
 Hausdorff
measure and dimension
 The Lebesgue
differentiation theorem
 Motivation from the
Fundamental Theorem of Calculus (Part
I) and the RadonNikodym Theorem
 The HardyLittlewood maximal function
 Proof of the Lebesgue differentiation
theorem
 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
 Monotone
rightcontinuous functions are
differentiable almost everywhere (using
the associated LebesgueStieltjes
measure, RadonNikodym and Lebesgue
differentiation theorems)
 Discussion of the
notion of absolute continuity
 The Fundamental
Theorem of Calculus (Part II)
