If X is a Banach Space, because it is a complete metric space Baire category theorem
applies. If X = Ci is a countable union of closed sets at least one of them will have
i=1
interior. If cfw_Ci are not closed then at lest one of them will have a closure
Let C (X ) be the space of continuous functions on a compact metric space X . Let
(f ) be a non-negative linear functional on C (X ). Then there is a measure on the Borel
sets B of X such that
(f ) =
f d
for all f C (X ).
We dene for closed sets C X
(C )
The completion of rationals. Let Q be the set of rational numbers. Consider the
set R of all partitions of Q into two nonempty disjoint sets L and R, with the following
properties. Such a partition will be called a Dedekind cut.
1. Q = L R
2. L R =
3. Fo
One of the goals of the course is to learn how to present proofs in a logically precise
and easily understandable form. Answers submitted in sloppy form will be returned even
if correct!
Example: Page 39 problem 6.
lim sup xn + lim inf yn lim sup(xn + yn
Assignment 9.
Problem 1. Denote by f =
0 f (x) 1 a.e. Dene
d
d .
Then since (A) (A) for all A , we have
A = cfw_ x : f ( x) = 0 , B = cfw_ x : 0 < f ( x) < 1 , C = cfw_ x : f ( x) = 1
Since (A) = A f (x)d and f (x) = 0 a.e. on A, clearly (A) = 0. Since
Real Variables Fall 2007.
Assignment 6. Due Oct 15.
Problem 1. We have a measure space (X, F , ) which is not a nite set of points. More
precisely assume that there is a countable collection of disjoint measurable sets cfw_Aj such
that 0 < (Aj ) < for ea
Real Variables Fall 2007.
Assignment 7. Due Oct 22.
Problem 1. cfw_fn is a sequence of integrable non-negative functions on a nite measure
space and fn f almost everywhere. Moreover
lim
n
fn d =
f d
so that equality holds in Fatous lemma. Show that cfw_f
Countable product measures.
Let X =
Xi be a product space. X x = cfw_xj where xj Xj . j is a -eld of
subsets of Xj and j is a countably additive measure on (Xj , j ) with (Xj ) = 1 for
every j . A nite dimensional cylinder set is a set of the form A = B
Real Variables Fall 2007.
Assignment 8. Due Oct 29.
Problem. Let cfw_xj : j = 1, 2, . . . be a countable set of points in [0, 1] and cfw_pj a set of
nonnegative numbers with s = j pj < . Dene
F ( x) =
pj
j :xj x
Of course F (x) 0 for x < 0 and F (x) s fo
Assignment 8.
F ( x) =
pj
j :xj x
F (x + h) F (x)
> q
h
Aq = cfw_x : lim sup
h0
We can exclude from A the set cfw_xj which is only countable. For each x Aq given any
> 0, there exists h < such that
F (x + h) F (x) qh
Since F (x) is right continuous, one
Real Variables Fall 2007.
Assignment 11. Due Nov 19.
Problem 1. Let f (x) be a continuous function of x dened on 0 x 1. Show that the
following sequence pn (x) of polynomials of degree n converges to f (x) uniformly on [0, 1]
as n .
n
j
n
f ( )xj (1 x)nj
Answers to Assignment 10.
Problem 1. d(x, A) = 0 if and only if there exists yn A such that d(x, yn ) 0, i.e., if
and only if x A or if A is closed if and only if x A.
Problem 2. If A X and xn is Cauchy in A, it is Cauchy in X and if X is complete
converg
Homeworks: These will be posted on the website before class on most Wednesdays. They are due back to me on the following Wednesday, and will be returned to you
on the Wednesday after that. If you can't make it to class, then you can give the homework to m