Real Analysis HW 1 Solutions
Problem 1: Prove that if A and B are two sets in A with A B, then m(A) m(B).
Proof: Since A B we can split up B into a union of two disjoint sets B = A (B A).
Using the countable additivity of m we find
m(B) = m (A (B A) = m(A
Real Analysis HW 12 Solutions
Problem 11: For a point x = (x1 , x2 , . . . , xn ) in Rn , define Tx to be the step function on
the interval [1, n + 1), that takes the value xk on the interval [k, k + 1), for 1 k n. For
p 1, define |x|p = |Tx |p , the norm
Real Analysis qual study guide
James C. Hateley
1. Measure Theory
Exercise 1.1. If A R and > 0 show open sets O R such that m (O) m (A) + .
[
Proof: Let cfw_In be a countable cover for A, then A
In . Since m (O) m (A) + . This implies
n=1
that
(
m (O) m
Real Analysis HW 7 Solutions
Problem 44: Let f be integrable over R and > 0. Establish the following three approximation properties.
R
(i) There is a simple function on R which has finite support and R |f | <
(ii) RThere is a step function s on R which v
Real Analysis HW 10 Solutions
Problem 47: Show that a function f is absolutely continuous on [a, b] if and only if for
each > 0, there is a > 0 such that for every finite disjoint collection cfw_(ak , bk )nk=1 of open
intervals in (a, b),
n
n
X
X
[bk ak ]
Real Analysis HW 5 Solutions
Problem 7: Let f be an increasing real-valued function on [0, 1]. For a natural number
n, define Pn to be the partition of [0, 1] into n subintervals on length 1/n. Show that
U (f, Pn ) L(f, Pn ) 1/n[f (1) f (0)]. Use Problem
Real Analysis HW 9 Solutions
Problem 33: Let cfw_fn be a sequence of functions on [a, b] converging pointwise to f . Then
T V (f ) lim inf n T V (fn ).
Solution: Let P be any partition of [a, b], then since V (fn , P ) only depends on fn at a finite
numb
Real Analysis HW 11 Solutions
Problem 1: For f in C[a, b], define
b
Z
kf k1 =
|f |.
a
Show that this is a norm on C[a, b]. Also show that there is no number c 0 for which
kf kmax ckf k1 for all f in C[a, b],
but there is a c 0 for which
kf k1 ckf kmax for
Real Analysis HW 8 Solutions
Problem 6: Let cfw_fn f in measure on E and g be a measurable function on E that is
finite a.e. on E. Show that cfw_fn g in measure on E if and only if f = g a.e. on E.
Solution: Suppose that cfw_fn g in measure, then there
Real Analysis HW 2 Solutions
Problem 18: Let E have finite outer measure. Show that there is a G set G E with
m(G) = m (E). Show that E is measurable if and only if there is an F set F E with
m(F ) = m (E).
Solution: Let n be a positive integer. By the de
Real Analysis HW 6 Solutions
Problem 33: Let cfw_fn be a sequence of integrable functions
R on E for which fn converges
to f a.e.R on E and
f
is
integrable
over
E.
Show
that
|f fn | 0 if and only if
E
R
limn E |fn | = E |f |.
R
Solution: If E |f fn | 0 t
Real Analysis HW 4 Solutions
Problem 9: Let cfw_fn be a sequence of measurable functions defined on a measurable set
E. Define E0 to be the set of points x in E at which cfw_fn (x) converges. Is the set E0
measurable.
Solution: Note that we may write E0
Real Analysis HW 3 Solutions
Problem 31. Justify the assertion in the proof of Vitalis Theorem that it suffices to
consider the case that E is bounded.
Solution. We know by a previous problem that any positive measure set contains a bounded
set which is a
Real Analysis HW 13 Solutions
Problem 1: Let X be a normed linear space and let T be a linear functional on X. Define
the norm
kT k = infcfw_M : |T (f )| M kf k for all f X.
Show that
kT k = supcfw_T (f ) : f X, kf k 1.
Solution: Define M = supcfw_ T (f )