MEASURES ON TOPOLOGICAL SPACES
PROBLEMS
Week 1.
1. Define the Cantor set C ⊂ [0, 1] as follows. Let C0 = [0, 1] and let Cn (n ≥ 1
be obtained from Cn−1 by removing the middle third open interval from each
T
of the 2n−1 intervals constituting it. Then put C = n∈N Cn . Show that C
is a null set for the Lebesgue measure but has uncountably many points.
[Hint: Use ’trinary’ expansion.]
2. A δ-ring is a ring R with the additional property that if An ∈ R (n ∈ N),
T
then n∈N An ∈ R.
Show that
τ (R) := {A ⊂ Ω : A ∩ B ∈ R ∀B ∈ R}
is a σ-algebra, and if µ is a measure on R (i.e. σ-additive), it can be extended
to a measure µ̃ on τ (R) by
µ̃(A) =
sup
B⊂A;B∈R
1
µ(B).
Week 2.
1. Assume that the Lebesgue measure is well-defined on the Borel σ-algebra, i.e.
the smallest σ-algebra containing all intervals, and satisfying mLeb (I) = |I|
for intervals I. Show that a continuous function f : [a, b] → C on a closed
bounded interval [a, b] and extended by 0 outside, is integrable, and that for
such functions
Z
Z
b
f dmLeb = (R)
f (x) dx.
a
2. Prove Fatou’s Lemma: If (fn )n ∈ N is a sequence of non-negative measurable
functions, then
Z
Z
lim inf fn dµ ≤ lim inf
n→∞
n→∞
fn dµ.
3. (i) Give an example of a monotonically increasing sequence of functions fn ≥ 0
R
R
on R such that fn (x)dx < +∞ for all n, but limn→∞ fn (x)dx = +∞.
(ii) Consider the functions fn on R defined by
fn (x) =
Show that
R
½ n−|x|
0
n2
fn (x)dx = 1 for all n, but
R
if |x| ≤ n, .
if |x| > n.
limn→∞ fn (x)dx = 0. Argue that this
agrees with Fatou’s lemma, and that the Dominated Convergence Theorem
does not apply.
2
Week 3.
1. Suppose that (Xi )i∈I is a collection of subspaces of a topological space X .
Q
The product space i∈I Xi is defined as follows:
Y
Xi = {f : I → X : f (i) ∈ Xi ∀i ∈ I},
i∈I
with open sets defined as follows: O ⊂
Q
i∈I
Xi is open if for every function
f ∈ O there exist a finite number of indices i1 , . . . , in ∈ I and open sets
Uip ⊂ Xip (p = 1, . . . , n) such that f (ip ) ∈ Uip for p = 1, . . . , n and
{g : g(ip ) ∈ Uip ∀p = 1, . . . , n} ⊂ O.
This called the product topology.
(i) Show that if Xi is Hausdorff for every i ∈ I (esp. if X is Hausdorff) then
Q
i∈I Xi is also Hausdorff.
Q
(ii) Let Xi = [0, 1] for all i ∈ I. Show that i∈I Xi = [0, 1]I is metrizable if
and only if I is countable.
[Hint. If I is uncountable, show that [0, 1]I has no countable neighbourhood
base.]
2. The Sorgenfrey line R[) is the real line but with topology generated by the halfopen intervals [a, b). Show that this is a Hausdorff topology, and determine
the compact sets of R[) .
3
Week 4.
1. The Dirac delta-measure at a point x of an arbitrary Hausdorff space X is
defined by
½
1 if x ∈ A;
δx (A) =
.
0 if x ∈
/ A;
for A ∈ B(X ). Show that this is a Radon measure on any Hausdorff space
X . Now consider the real line with the discrete topology, i.e. the topology
consisting of all subsets of R. Show that an arbitrary Radon measure on Rdisc
is of the form
X
µn δx
x∈D
where µn > 0 and D is a denumerable subset if mu is moderate. What is the
corresponding support? When is a measure of this form a Radon measure on
R with the usual (Euclidean) topology? What is its support?
2. A Radon probability measure on X is a Radon measure with µ(X ) = 1.
Given a Radon probability measure µ on R, define its distribution function
F by F (x) = µ((−∞, x]). Show that F satisfies:
1. F is non-decreasing;
2. F is right-continuous, i.e. if xn ↓ x then F (xn ) ↓ F (x);
3. limx→−∞ F (x) = 0 and limx→+∞ F (x) = 1.
Conversely, given a function F : R → [0, 1] with these properties, prove that
there exists a unique Radon probability measure for which it is the distribution
function.
[Hint: An open subset of R is a countable union of open intervals.]
4
Week 5.
1. Prove the following theorem:
Theorem. Let (Ω, A, µ) be a measure space. Suppose that f : (a, b) × Ω → C
is a function of two variables t ∈ (a, b) ⊂ R and x ∈ Ω such that f (t, ·) ∈
L1 (Ω). Define
Z
F (t) = f (t, x) µ(dx).
Suppose, moreover, that ∂f
∂t (t, x) exists for all t ∈ (a, b) and a.e. x ∈ Ω, and
that there exists a function h ∈ L1 (Ω) such that
¯
¯
¯
¯ ∂f
¯ (t, x)¯ ≤ h(x)
¯
¯ ∂t
for all t ∈ (a, b) and a.e. x ∈ Ω. Then F (t) is differentiable on (a, b) and
dF (t)
=
dt
Z
∂f
(t, x)µ(dx).
∂t
2. Consider the function
f (t, x) = e−tx
sin(x)
x
for t > 0, x ≥ 0. Show that the above theorem applies and compute the
derivative of
Z ∞
F (t) =
f (t, x) dx.
0
Hence determine F (t). Now define, for n ∈ N,
Z
n
Fn (t) =
f (t, x) dx.
0
Show that Fn (t) is continuous on (0, +∞) and that the limit limt→0 Fn (t)
exists. Show also that Fn converges uniformly to F .
Conclude that limt→0 F (t) = F (0), and compute the limit.
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Week 6.
1. Determine the integral
Z
1R (x, y)x4 y cos(y 3 ) dx dy
where R is the region
R = {(x, y) ∈ R2 : −1 ≤ x ≤ 1, −1 ≤ y ≤ x5 }.
2. (i) Let µ be a Radon probability measure on a Hausdorff space X , and f :
X → Y a continuous map to a Hausdorff space Y. Show that the image
measure f (µ) defined by
f (µ)(A) = µ(f −1 (A))
is a Radon probability measure on Y.
(ii) Given a Radon probability measure ν on R and a function f : R → [0, ∞)
which is increasing for x ≥ a, prove Chebyshev’s inequality:
1
ν ([a, +∞)) ≤
f (a)
Z
f (x) ν(dx)
provided f (a) > 0.
(iii) Let µ be a Radon probability measure on R with finite variance:
R 2
x µ(dx) < +∞. Consider the sequence (µn )∞
n=1 of probability measures µn
n
on R defined by µ1 = µ and µn+1 = µn × µ. Define the measures νn on R
by νn = gn (µn ), where gn : Rn → R is defined by
n
1X
gn (x1 , . . . , xn ) =
xi .
n i=1
Show that for any ² > 0,
lim νn ((m − ², m + ²)) = 1,
n→∞
where m =
R
x µ(dx).
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Week 7.
1. Let µ be a probability measure on Rn and define the map π : Rn → [0, +∞)
Pn
by π(x) = |x|, where |x|2 = i=1 x2i . Let ν = π(µ) be the image measure (See
Problem 2 of Week 6.). Prove that there exists a conditional expectation
map E : L1 (Rn , µ) → L1 ([0, +∞), ν) satisfying
Z
Z
E(f ) dν =
f dµ
(∗)
π −1 (A)
A
for all A ∈ B([0, +∞)), as follows:
(i) First define E : L2 (Rn , µ) → L2 ([0, +∞), ν) by
hE(f ) | gi = hf | π̃(g)i
where π̃(g) = g ◦ π. (This uses Riesz’ Lemma, which states that an element
ξ of a Hilbert space H is uniquely determined by its scalar products hξ | ηi
with every η ∈ H.) Show that the map E thus defined satisfies (*) for f ∈
L2 (Rn , µ).
(ii) Then extend E to L1 by first proving that it is a positive map, i.e.
f ≥ 0 =⇒ E(f ) ≥ 0,
and then deducing that
|E(f )| ≤ E(|f |)
first for real-valued f and then for complex-valued f by writing |E(f )| =
eiθ E(f ).
2. The n-dimensional normal distribution with variance σ 2 is the measure on Rn
defined by
Z
2
2
dn x
γσ (A) =
e−|x| /(2σ )
.
(2πσ 2 )n/2
A
(i)
Show that
Z
R
(ii)
n
|x|2 µ(dx) = σ 2 .
Argue that the measure ν defined in the first problem is given by
Z
2
2
rn−1 dr
n−1
,
ν(A) = |S
|
e−r /(2σ )
(2πσ 2 )n/2
A
where |S n−1 | is the (n − 1)-dimensional Lebesgue measure of the unit sphere
S n−1 = {x ∈ Rn : |x| = 1}.
(iii)
Now apply (*) to the function f (x) = |x|2 to determine |S n−1 |.
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Week 8.
1. Prove that there exists no nontrivial translation-invariant Radon measure on
an infinite-dimensional Hilbert space.
2. Show that a subset K of a separable Hilbert space H is compact if and only if
it is closed, bounded, and uniformly truncatable in a given orthonormal basis
{en }n∈N , i.e. for all ² > 0, there exists N ∈ N such that for all x ∈ K,
||x − PN x|| < ², where Pn is the projection onto the span of e1 , . . . , eN .
3.∗ Let H be a (separable) Hilbert space. A projective system of measures on
H is a collection of Radon measures µF for all finite-dimensional subspaces F
of H such that for all finite-dimensional subspaces F1 ⊂ F2 , µF1 = πF2 ,F1 (µF2 )
is the image measure under the projection map πF2 ,F1 : F2 → F1 . Let (µF )F
be a projective system of probability measures on H. Prove that there exists
a unique Radon probability measure µ on H such that πF (µ) = µF for all
finite-dimensional subspaces F (where πF : H → F is the projection onto F )
if and only if for all ² > 0 there exists a compact set K ⊂ H (independent of
F ) such that
µF (πF (K)) > 1 − ²
for all finite-dimensional subspaces F .
[Hint. Choose an orthonormal basis for H and use the Riesz-Markov theorem.
Use also the fact that every function f ∈ κ(H) is uniformly continuous.]
4. Given a separable real Hilbert space H, and a bounded symmetric operator
T ∈ L(H) with strictly positive eigenvalues, define Gaussian measures γF on
the finite-dimensional subspaces as follows: If {e1 , . . . , en } is an orthonormal
Pn
basis of F , define a map uF : Rn → F by u(x1 , . . . , xn ) = i=1 xi ei , and let
TF = (hei , T ej i) i, j = 1n be the matrix of T restricted to F on this basis.
Then, for a Borel subset A ⊂ F ,


n
X
1
dx1 . . . dxn
γF (A) =
exp −
hei , T ej ixi xj 
.
−1
2 i,j=1
(2π)n/2 (det TF )1/2
uF (A)
Z
Show that this defines a projective system of probability measures on H. Show
also, that if T is the identity map, then there does not exists a measure γ on
H such that γF = πF (γ).
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