MATH 516 Supplementary Homework
Spring 2015
1. Let X denote the set of integers, and define
τ = {E ⊂ X : E c is finite} ∪ ∅
a) Show that τ is a topology on X (the so-called co-finite topology).
b) Show that (X, τ ) is a Tychonoff space, but not a Hausdorff space.
c) Are limits of sequences unique in (X, τ )?
2. If ν is a signed measure, find the Radon-Nikodym derivative
dν
.
d|ν|
3. Let X = Z+ = {1, 2, . . . }, an , bn > 0 for all n ∈ Z+ and
X
X
µ(E) =
an
ν(E) =
bn
n∈E
n∈E
(These are ’weighted’ counting measures.) Show that ν << µ, µ << ν and find the
dν dµ
Radon-Nikodym derivatives
, .
dµ dν
4. Let (X, M, µ) be a measure space and 1 ≤ p < q < ∞.
a) Suppose that there exists > 0 such that µ(E) ≥ for any nonempty E ∈ M.
Show that Lp (µ) ⊂ Lq (µ). (Suggestion: first show Lp (µ) ⊂ L∞ (µ).)
b) If M contains sets of arbitrarily small positive measure, show that Lp (µ) is not
contained in Lq (µ). (Suggestion: find disjoint
sets En ∈ M such that 0 < µ(En ) < 2−n
P
and consider functions of the form f =
an χEn .)
5. Suppose µ(X) = 1 and f ∈ Lp for some p > 0, so that f ∈ Lq for 0 < q < p. Show
that
R
a) log ||f ||q ≥ X log |f | dµ
R
b) 1q ( X |f |q dµ − 1) ≥ log ||f ||q
R
R
c) 1q ( X |f |q dµ − 1) → X log |f | dµ as q → 0+
d) limq→0+ ||f ||q = exp (
R
X
log |f | dµ) (with the convention that exp (−∞) = 0).
(Jensen’s inequality may be helpful for some of this.)
6. Let (X, M, µ) be a σ-finite measure space, let f be a non-negative measurable function
on X, and
R = {(x, y) ∈ X × [0, ∞) : y < f (x)}
a) Show that R is measurable with respect to the product measure µ × m on X × [0, ∞).
(Suggestion: first suppose f is a simple function.)
R
b) Show that (µ × m)(R) = X f dµ.
c) The graph of f is by definition
G = {(x, y) ∈ X × [0, ∞) : y = f (x)}
If f ∈ L1 (X), show that G has µ × m measure zero.
7. If µ is a Radon
measure on a locally compact Hausdorff space X, φ ∈ L1 (µ), φ ≥ 0
R
and ν(E) = E φ dµ, show that ν is a also a Radon measure on X.
8. Let X be a normed vector space and M ⊂ X a proper closed subspace of X. Define
the equivalence relation x ∼ y if x − y ∈ M . The equivalence class of x is denoted by
x + M , and the set of all equivalence classes is the quotient space X/M .
a) Verify that X/M is a vector space with the operations
(x + M ) + (y + M ) = (x + y) + M
λ(x + M ) = (λx) + M
b) Show that
||x + M || = inf{||x + y|| : y ∈ M }
defines a norm on X/M .
c) Show that for any > 0 there exists x ∈ X such that ||x|| = 1 and ||x+M || ≥ 1−.
d) Define the projection map π : X → X/M by π(x) = x + M , and show that
||π|| = 1.
e) If X is a Banach space, show that X/M is also a Banach space.
9. Let X = R2 , S = {(x, y) ∈ R2 : y = 0}, and define f : S → R by f (x, 0) = x. Thus
clearly ||f || = 1 considered as a linear functional on S.
a) Show that there is only one extension F of f to all of X such that ||F || = 1,
provided we use the ordinary Euclidean distance on R2 .
b) Show that there are infinitely many such extensions F for which ||F || ≤ 1.01.
c) If instead we use the 1 norm on R2 , ||(x, y)|| = |x| + |y|, show that there are
infinitely many extensions of F of f such that ||F || = 1.