Math 202B UCB, Spring 2013 M. Christ
Lecture 4, Wednesday January 30, 2013
Administrative announcements:
Solutions to problem set 1 will be posted today (Wednesday).
Please continue reading last two sections of Chapter V.
Integration of C and vectorvalu
Lecture 32, Monday April 15
We concluded on Friday with the statement of the Riesz Representation Theorem for
Hilbert spaces. Every element of H is of the form z (x) = x, z for some z H; the
correspondence z z between H and H is norm-preserving. Before gi
Math 202B UCB, Spring 2013 M. Christ
Lecture 35, Monday April 22
On Uniqueness
On Friday we stated the Hahn-Banach Theorem and discussed two corollaries. The
second stated that if X is a Banach space and x X , there exists X satisfying
is unique; (y ) =
X
Math 202B UCB, Spring 2013 M. Christ
Lecture 38, Monday April 29 (and ensuing lectures)
Fourier Series
The Setting
Dene T = cfw_z C : |z | = 1. This is a compact metric space. It is often parametrized
by x eix , with x [, ]; this parametrization is not qu
Math 202B UCB, Spring 2013 M. Christ
Problem Set 1, Due Wednesday January 30
(In all problems, you may take for granted Proposition VI.1.)
(1.1) Let X be an uncountable set. Let A P (X ) be the algebra consisting of all sets
which are either countable, or
Math 202B UCB, Spring 2013 M. Christ
Problem Set 2, Due Wednesday February 6
Throughout, (X, A, ) denotes a measure space.
(2.1) Let an be nonnegative numbers satisfying an = . Construct an example of a
n=1
sequence of measurable functions on some measure
Math 202B UCB, Spring 2013 M. Christ
Problem Set 3, Due Wednesday February 13
Final Version
While lectures go on about generalities, this problem set continues to build skills relevant
to more concrete problems which arise in analytic practice.
Throughout
Math 202B UCB, Spring 2013 M. Christ
Problem Set 4, due Wednesday 2/27 (Final version)
Please read VI.8, and the rst two pages (only) of VI.9, which concern the denition of
the class of functions of bounded variation. Skip over the rest of Chapter VI (but
Math 202B UCB, Spring 2013 M. Christ
Problem Set 5, due Wednesday 3/6 (preliminary version)
(5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on
no interval of positive length. (Hint: Use the fact, proved last semester, th
Math 202B UCB, Spring 2013 M. Christ
Problem Set 6, due Friday 3/22
(6.1) Let (X, A, ) and (Y, B , ) be two nite measure spaces. Let f : X Y R be a measurable
function. Let fx (y ) = f (x, y ). For x X dene F (x) = fx L (Y ) for all x for which fx is a
me
Math 202B UCB, Spring 2013 M. Christ
Problem Set 7, due Friday 4/5
(7.1) Let X be a compact Hausdor space and let be a (nonnegative) Radon measure on X . Show
that for any Borel measurable function f : X C and any > 0 there exists a compact set K X
such t
Math 202B UCB, Spring 2013 M. Christ
Problem Set 8, due Wednesday 4/10/2013
(8.1) Let be a complex measure on (X, A). The associated (positive) measure | is dened by
|(E ) = sup
|(Ej )|
j
for E A, where the supremum is taken over all decompositions of E a
Math 202B UCB, Spring 2013 M. Christ
Problem Set 9, due Wednesday 4/17/2013
An inner product space (H, , ) is a real or complex vector space H equipped with an inner
product satisfying the usual properties. x = x, x 1/2 will denote the associated norm. A
Math 202B UCB, Spring 2013 M. Christ
Problem Set 10, due Wednesday 4/24/2013
(10.1) Generalize problem (7.6) to C k ([0, 1]d ) for any dimension d cfw_1, 2, 3, .
(10.2) A normed linear linear space is said to be separable if it is separable as a metric sp
Lecture 29, Monday 4/8/2013
In the rst sentence of todays lecture, I will complete the proof of the second Riesz
Representation Theorem. See the conclusion of the notes posted for Friday 4/5.
Please also note the addition to those lecture notes of a brief
Lecture 26, Monday 4/1/2013
And ensuing lectures
Where were we?
1. We stated the two Riesz Representation Theorems, concerned respectively with positive linear functionals on Cc (X ), and with bounded linear functionals on C0 (X ),
where X is any locally
Math 202B UCB, Spring 2013 M. Christ
Lectures 24 and 25, Wednesday-Friday 3/2022/2013
Integration on Locally Compact Hausdor Spaces
and (especially) the Riesz Representation Theorems
For this topic I will follow the treatment in G. Follands book Real Anal
Math 202B UCB, Spring 2013 M. Christ
Lecture 6, Monday February 4 or thereabouts, 2013
Administrative announcements:
Today we will deviate from V.9 to discuss the proofs, for general exponents p, of key
inequalities which are proved in that section of ou
Math 202B UCB, Spring 2013 M. Christ
Lecture 7, Wednesday 2/6/2013
Administrative announcements:
Please VI.1.
Regularity of Borel Measures
Denition. Borel measure: A (nonnegative) measure on B = Bn = B (Rn ) which satises
(K ) < for every compact set K R
Math 202B UCB, Spring 2013 M. Christ
Lecture 9, Monday 2/11/2013
Administrative announcements:
Please read VI.3, VI.4 if you have not already done so. III.8 is also relevant to the
discussion of the connection between Riemann and Lebesgue dierentiation.
Math 202B UCB, Spring 2013 M. Christ
Lecture 11, Friday 2/15/2013
Change of variables in Lebesgue integrals
A useful fact: By a cube in Rn I will (during the discussion of change of variables) mean
a product n [ai , bi ) where bi ai > 0 is independent of
Math 202B UCB, Spring 2013 M. Christ
Lecture 12, Wednesday 2/20/2013
Change of variables in Lebesgue integrals (conclusion)
A general remark: If is a measure, and if h : X [0, ] is a measurable function,
dene a measure by (E ) = E h d for E A. This is ind
Math 202B UCB, Spring 2013 M. Christ
Lecture 13, Friday 2/22/2013
Stieltjes Measures, and Monotonic Functions
Some elementary properties of nondecreasing functions f : R R (thus x y
f (x) f (y ): For each x R, the two one-sided limits limtx f (t) both ex
Math 202B UCB, Spring 2013 M. Christ
Lecture 14, Monday 2/25/2013
Stieltjes Measures, and Monotonic Functions (continued)
Last time we constructed the Cantor-Lebesgue function, F . This function is continuous
and nondecreasing, yet its derivative exists a
Math 202B UCB, Spring 2013 M. Christ
Lecture 15, Wednesday 2/27/2013
(a.e.) Dierentiability of Monotonic Functions
Theorem Let f be a bounded real-valued function on some interval. Then f is of bounded
variation if and only if f can be expressed as a dier
Math 202B UCB, Spring 2013 M. Christ
Lecture 16, Friday 3/1/2013
(a.e.) Dierentiability of Monotonic Functions
Here is a consequence of the Rising Sun Lemma which begins to indicate how the RSL
is connected with dierentiability questions.
Lemma. Let F : [
Math 202B UCB, Spring 2013 M. Christ
Lecture 17, Monday 3/4/2013
(a.e.) Dierentiability of Monotonic Functions (part 2)
Today we prove:
Theorem. Let F : [a, b] R be continuous and nondecreasing. Then F (x) exists for
almost every x (a, b).
Recall that any
Math 202B UCB, Spring 2013 M. Christ
Lectures 18/19, Wednesday/Friday 3/6 and 3/8/2013
Signed Measures, Hahn/Jordan Decomposition, Radon-Nikodym Theorem
Our midterm exam is next Wednesday, 3/13. It will be based on all material discussed
in the course thr
Math 202B UCB, Spring 2013 M. Christ
Lecture 20, Monday 3/12/2013
Radon-Nikodym Theorem (conclusion)
Our midterm exam is this Wednesday, 3/13. It will be based on all material discussed
in the course through the end of last week.
Solutions to the rst 5 pr
Math 202B UCB, Spring 2013 M. Christ
Lectures 22 and 23, Friday 3/15 and Monday 3/18/2013
(Lecture 21, Wednesday 3/13, was the midterm exam.)
Theorem. Let M be the maximal function, as dened in Lecture 20. Then for all f L1 (R)
and all t > 0,
m cfw_x : M