Solutions to some exercises on outer measure
#21, p. 352. Well show that (E ) = 1 for every nonempty subset E of X . To see this, observe
that the only collection of sets in S which covers E (ignoring the empty set, which has no eect), is the
collection c
Math 5463 Spring 2013
Exam 2
(Note: For these problems, you may cite any result that we have done in class, without having to prove
it.)
1. Suppose cfw_n nN is an orthonormal basis of a Hilbert space H .
a) Show that m m = 2 for all natural numbers m and
Math 5463 Review for Exam 1
Exam 1 covers chapter 7 of Royden and Fitzpatricks test.
7.1. Weve covered everything in this section.
7.2. We covered everything in this section up through the Cauchy-Schwarz inequality on page 142. However,
I gave dierent pro
Comment on second problem on Assignment 5
The second problem on Assignment 5 considered the situation in which a function f was measurable
with respect to a -algebra on a set X , but not necessarily measurable with respect to another, smaller,
-algebra 0
Math 5463 Spring 2013
Final Exam
1. Suppose (X, ) is a measure space and fn are functions in Lp (X, ) which converge pointwise to a
function f . Suppose also that there exists a constant M < such that fn M for all n N. Show
that f is in Lp (X, ).
2. Suppo
Math 5463 Review for nal
The material weve covered in class corresponds, more or less, to the following sections of Royden and
Fitzpatrick:
7.1, 7.2, 7.3, 7.4, 13.1, 13.2, 16.1, 16.3, 17.1, 17.2, 17.3, 17.4, 17.5, 18.1, 18.2, 18.3, 18.4, 19.2, 20.1;
or th
1. (problem #7, page 312 of Royden and Fitzpatrick) Suppose X is a vector space over the real numbers,
and is a normed space, with norm . We say that the norm is induced by an inner product if (i) there
exists an inner product dened on X , that is, for al