Contents 1. σ-algebras 2 1.1. The Borel σ-algebra over R 5 1.2. Product σ-algebras 7 2. Measures ...

96 Pages·2015·0.68 MB·English

Contents
1. σ-algebras 2
1.1. The Borel σ-algebra over R 5
1.2. Product σ-algebras 7
2. Measures 8
3. Outer measures and the Caratheodory Extension Theorem 11
4. Construction of Lebesgue measure 15
5. Premeasures and the Hahn-Kolmogorov Theorem 21
6. Lebesgue-Stieltjes measures on R T 24
7. Problems 28
8. Measurable functions 32
F
9. Integration of simple functions 38
10. Integration of unsigned functions 40
11. Integration of signed and complex funcAtions 44
11.1. Basic properties of the absolutely convergent integral 45
12. Modes of convergence 48
R
12.1. The five modes of convergence 48
12.2. Finite measure spaces 53
12.3. Uniform integrability 54
D
13. Problems 57
13.1. Measurable functions 57
13.2. The unsigned integral 58
13.3. The signed integral 59
13.4. Modes of convergence 59
14. The Riesz-Markov Representation Theorem 61
14.1. Urysohn’s Lemma and partitions of unity 62
14.2. Proof of Theorem 14.2 62
15. Product measures 66
16. Integration in Rn 74
17. Differentiation theorems 78
18. Signed measures and the Lebesgue-Radon-Nikodym Theorem 82
18.1. Signed measures; the Hahn and Jordan decomposition theorems 83
18.2. The Lebesgue-Radon-Nikodym theorem 87
18.3. Lebesgue differentiation revisited 90
1
19. Problems 91
19.1. Product measures 91
19.2. Integration on Rn 91
19.3. Differentiation theorems 92
19.4. Signed measures and the Lebesgue-Radon-Nikodym theorem 93
19.5. The Riesz-Markov Theorem 94
Index 95
T
MAA6616 COURSE NOTES
FALL 2015
F
1. σ-algebras
A
Let X be a set, and let 2X denote the set of all subsets of X. Let Ec denote the
complement of E in X, and for E,F ⊂ X, write E \F = E ∩Fc.
Definition 1.1. Let X be a set. A Boolean algebra is a nonempty collection A ⊂ 2X
R
which is closed under finite unions and complements. A σ-algebra is a Boolean algebra
which is also closed under countable unions.
If M ⊂ N ⊂ 2X are σ-algebras, then M is coarser than N . Likewise N is finer
D
than M. (cid:47)
Remark 1.2. If E is any collection of sets in X, then
α
(cid:32) (cid:33)c
(cid:91) (cid:92)
Ec = E . (1)
α α
α α
Hence a Boolean algebra (resp. σ-algebra) is automatically closed under finite (resp.
countable) intersections. It follows that a Boolean algebra (and a σ-algebra) on X
always contains ∅ and X. (Proof: X = E ∪Ec and ∅ = E ∩Ec.) (cid:5)
Definition 1.3. A measurable space is a pair (X,M) where M ⊂ 2X is a σ-algebra. A
function f : X → Y from one measurable space (X,M) to another (Y,N ) is measurable
if f−1(E) ∈ M whenever E ∈ N . (cid:47)
Definition 1.4. A topological space X = (X,τ) consists of a set X and a subset τ of
2X such that
(i) ∅,X ∈ τ;
(ii) τ is closed under finite intersections;
(iii) τ is closed under arbitrary unions.
Date: November 20, 2015.
2
MAA6616 COURSE NOTES FALL 2015 3
The set τ is a topology on X.
(a) Elements of τ are open sets;
(b) A subset S of X is closed if X \S is open;
(c) S is a G if S = ∩∞ O for open sets O ;
δ j=1 j j
(d) S is an F if it is an (at most) countable union of closed sets;
σ
(e) A subset C of X is compact, if for any collection F ⊂ τ such that C ⊂ ∪{T : T ∈ F}
there exist a finite subset G ⊂ F such that C ⊂ ∪{T : T ∈ G};
(f) If τ and σ are both topologies on X, then τ is finer than σ (and σ is coarser than
τ) if σ ⊂ τ; and
(g) If (X,τ) and (Y,σ) are topological spaces, a function f : X → Y is continuous if
S ∈ σ implies f−1(S) ∈ τ. T
(cid:47)
Example 1.5. If (X,d) is a metric space, then the Fcollection τ of open sets (in the
metric space sense) is a topology on X. There are important topologies in analysis that
are not metrizable (do not come from a metric). (cid:52)
A
Remark 1.6. There is a superficial resemblance between measurable spaces and topo-
logicalspacesandbetweenmeasurablefunctionsandcontinuousfunctions. Inparticular,
a topology on X is a collection of subsets of X closed under arbitrary unions and finite
intersections, whereas for a σ-algebrRa we insist only on countable unions, but require
complements also. For functions, recall that a function between topological spaces is
continuous if and only if pre-images of open sets are open. The definition of measur-
able function is plainly similar. The two categories are related by the Borel algebra
D
construction appearing later in these notes. (cid:5)
The disjointification trick in the next Proposition is often useful.
Proposition 1.7 (Disjointification). If M ⊂ 2X is a σ-algebra if and (G )∞ is a
j j=1
sequence of sets from M, then there exists a sequence (F )∞ of pairwise disjoint sets
j j=1
from M such that
n n
(cid:91) (cid:91)
F = G
j j
j=1 j=1
for n either a positive integer or ∞.
If ∅ (cid:54)= M ⊂ 2X is closed with respect to complements, finite intersections and
countable disjoint unions, then M is a σ-algebra. †
Proof. The proof amounts to the observation that if (G ) is a sequence of subsets of X,
n
then the sets
(cid:32) (cid:33)
n−1
(cid:91)
F = G \ G = G ∩(∩n−1Gc) (2)
n n k n k=1 k
k=1
are disjoint, in M and (cid:83)n F = (cid:83)n G for all n ∈ N+ (and thus (cid:83)∞ F = (cid:83)∞ G ).
j=1 j j=1 j j=1 j j=1 j
4 MAA6616 COURSE NOTES FALL 2015
To prove the second part of the Proposition, given a sequence (G ) from M use
n
the disjointification trick to obtain a sequence of disjoint sets F ∈ M such that ∪G =
n n
∪F . (cid:3)
n
Example 1.8. Let X be a nonempty set.
(a) The power set 2X is the finest σ-algebra on X.
(b) At the other extreme, the set {∅,X} is the coarsest σ-algebra on X.
(c) Let X be an uncountable set. The collection
M = {E ⊂ X : E is at most countable or X \E is at most countable } (3)
is a σ-algebra (the proof is left as an exercise).
(d) If M ⊂ 2X a σ-algebra, and E is any nonempty subset of XT, then
M := {A∩E : A ∈ M} ⊂ 2E
E
is a σ-algebra on E (exercise).
F
(e) If {M : α ∈ A} is a collection of σ-algebras on X, then their intersection ∩ M
α α∈A α
is also a σ-algebra (checking this statement is a simple exercise). Hence given any
set E ⊂ 2X, we can define the σ-algebra
A
(cid:92)
M(E) = {M : M is a σ-algebra and E ⊂ M}. (4)
Note that the intersection is over a nonempty collection since E is a subset of the
σ-algebra 2X. We call M(E) theRσ-algebra generated by E.
(f) An important instance of the construction in item (e) is when X is a topological
space and E is the collection of open sets of X. In this case the σ-algebra generated
by E is the Borel σ-algebra and is denoted B . The Borel σ-algebra over R is
D X
studied more closely in Subsection 1.1.
(g) If (Y,N ) is a measurable space and f : X → Y, then the collection
f−1(N ) = {f−1(E) : E ∈ N } ⊂ 2X (5)
is a σ-algebra on X (check this) called the pull-back σ-algebra. The pull-back σ-
algebra is the smallest σ-algebra on X for which the function f : X → Y is measur-
able.
(h) More generally given a family of measurable spaces (Y ,N ), where α ranges over
α α
some index set A, and functions f : X → Y , let
α α
E = {f−1(E ) : α ∈ A,E ∈ N }
α α α α
and letting M = M(E). Unlike the case of a single f, E by itself will not be a
σ-algebra in general. M is characterized by the property that it is the smallest
σ-algebra on X such that each of the functions f is measurable. An important
α
special case of this construction is the product σ-algebra (see Subsection 1.2).
(i) If (X,M) is a measurable space and f : X → Y, then
Ω = {E ⊂ Y : f−1(E) ∈ M}
f
is a σ-algebra.
(cid:52)
MAA6616 COURSE NOTES FALL 2015 5
The following proposition is trivial but useful.
Proposition 1.9. If M ⊂ 2X is a σ-algebra and E ⊂ M, then M(E) ⊂ M. †
The proposition is used in the following way. To prove a particular statement is
true for every set in some σ-algebra M (say, the Borel σ-algebra B ) generated by a
X
collection of sets E (say, the open sets of X), it suffices to prove that 1) the statement
is true for every set in E, and 2) the collection of sets for which the statement is true
forms a σ-algebra. (The monotone class lemma which we will study later is typically
used in a similar way.)
A function f : X → Y between topological spaces is said to be Borel measurable if
it is measurable when X and Y are equipped with their respectTive Borel σ-algebras.
Proposition 1.10. If X and Y are topological spaces and if f : X → Y is continuous,
then f is Borel measurable. †
F
Proof. Problem 7.6. (Hint: follow the strategy described after Proposition 1.9.) (cid:3)
A
1.1. The Borel σ-algebra over R. Before going further, we take a closer look at the
Borel σ-algebra over R, beginning with the following useful lemma on the structure of
open subsets of R which may be familiar to you from advanced calculus.
R
Lemma 1.11. Every nonempty open subset U ⊂ R is an (at most countable) disjoint
union of open intervals. †
Here we allow the “degDenerate” intervals (−∞,a),(a,+∞),(−∞,+∞).
Proof outline. First verify that if I and J are intervals and I ∩J (cid:54)= ∅, then I ∪J is an
interval. Given x ∈ U, let
α =sup{a : [x,a) ⊂ U}
x
β =inf{b : (b,x] ⊂ U}
x
and let I = (α ,β ). Verify that, for x,y ∈ U either I = I or I ∩ I = ∅. Indeed,
x x x x y x y
x ∼ y if I = I is an equivalence relation on U. Hence, U = ∪ I expresses U as
x y x∈U x
a disjoint union of nonempty intervals, say U = ∪ I where P is an index set and
p∈P p
the I are nonempty intervals. For each q ∈ Q ∩ U there exists a unique p such that
p q
q ∈ I . On the other hand, for each p ∈ P there is a q ∈ Q∩U such that q ∈ I . Thus,
pq p
the mapping from Q∩U to P defined by q (cid:55)→ p is onto. It follows that P is at most
q
countable. (cid:3)
Proposition 1.12 (Generators of B ). Each of the following collections of sets E ⊂ 2R
R
generates the Borel σ-algebra B :
R
(i) the open intervals E = {(a,b) : a,b ∈ R};
1
(ii) the closed intervals E = {[a,b] : a,b ∈ R};
2
(iii) the (left or right) half-open intervals E = {[a,b) : a,b ∈ R} or E = {(a,b] : a,b ∈
3 4
R};
6 MAA6616 COURSE NOTES FALL 2015
(iv) the (left or right) open rays E = {(−∞,a) : a ∈ R} or E = {(a,+∞) : a ∈ R};
5 6
(v) the (left or right) closed rays E = {(−∞,a] : a ∈ R} or E = {[a,+∞) : a ∈ R}.
7 8
†
Proof. Only the open and closed interval cases are proved, the rest are similar and left
as exercises. The proof makes repeated use of Proposition 1.9. Let O denote the open
subsets of R. Thus, by definition, B = M(O). To prove M(E ) = B , first note that
R 1 R
since each interval (a,b) is open and thus in O, M(E ) ⊂ M(O) by Proposition 1.9.
1
Conversely, each open set U ⊂ R is a countable union of open intervals, so M(E )
1
contains O and hence M(O) ⊂ M(E ).
1
For the closed intervals E , first note that each closed set iTs a Borel set, since it is
2
the complement of an open set. Thus E ⊂ B so M(E ) ⊂ B by Proposition 1.9.
2 R 2 R
Conversely, each open interval (a,b) is a countable union of closed intervals [a+1,b−1].
n n
Indeed, for −∞ < a < b < ∞,
F
∞
(cid:91) 1 1
(a,b) = [a+ ,b− ]
n n
n=N A
and a similar construction handles the cases that either a = −∞ or b = ∞. It follows
that E ⊂ M(E ), so by Proposition 1.9 and the first part of the proof,
1 2
BR =RM(E1) ⊂ M(E2).
(cid:3)
D
Sometimes it is convenient to use a more refined version of the above Proposition,
where we consider only dyadic intervals.
Definition 1.13. A dyadic interval is an interval of the form
(cid:18) (cid:21)
k k +1
I = , (6)
2n 2n
where k,n are integers. (cid:47)
(Draw a picture of a few of these to get the idea). A key property of dyadic intervals
is the nesting property: if I,J are dyadic intervals, then either they are disjoint, or one is
contained in the other. Dyadic intervals are often used to “discretize” analysis problems.
Proposition 1.14. Every open subset of R is a countable disjoint union of dyadic in-
tervals. †
Proof. Problem 7.4. (cid:3)
It follows (using the same idea as in the proof of Proposition 1.12) that the dyadic
intervalsgenerateB . Theuseofhalf-openintervalshereisonlyatechnicalconvenience,
R
to allow us to say “disjoint” in the above proposition instead of “almost disjoint.”
MAA6616 COURSE NOTES FALL 2015 7
1.2. Product σ-algebras. Suppose n ∈ N+ and (X ,M ) are σ-algebras for j =
j j
1,2,...,n. Let X = (cid:81)n X , the product space. Thus X = {(x ,...,x ) : x ∈
j=1 j 1 n j
X , j = 1,...,n}. Let π : X → X denote the j-th coordinate projection, π(x) = x .
j j j j
The product sigma algebra, defined below, is the smallest sigma algebra on X such that
each π is measurable.
j
Definition 1.15. If (X ,N ), j = 1,...n are measurable spaces, the product σ-algebra
j j
⊗n N is the σ algebra on X = (cid:81)n X generated by
j=1 j j=1 j
{π−1(E ) : E ∈ N ,j = 1,...n}.
j j j j
(cid:47)
T
There are now two canonical ways of constructing σ-algebras on Rn. The Borel
σ-algebra B and the product σ-algebra obtained by giving each copy of R the Borel
Rn
σ-algebra B and forming the product σ-algebra ⊗nBF. It is reasonable to suspect that
R 1 R
these two σ-algebras are the same, and indeed they are.
Proposition 1.16. B = ⊗n B . †
Rn j=1 R A
Proof. We use Proposition 1.9 to prove inclusions in both directions. By definition, the
product σ-algebra ⊗n B is generated by the collection of sets
k=1 R
R
E = {π−1(E ) : E ∈ B , j = 1,...n},
j j j R
where π (x ,...x ) = x is the projection map, π : Rn → R. By Example 1.8(i), for
j 1 n j
each 1 ≤ k ≤ n, D
M = {E ⊂ R : π−1(E) ⊂ B }
j j Rn
is a σ-algebra. For E ⊂ R, we have π−1(E ) = R×···×E ×···×R, where E is the jth
j j j j j
factor. Thus, if E is open, then π−1(E ) ∈ B . Consequently, M contains the open
j j j Rn j
sets and hence B . Thus, if E ∈ B , then π−1(E ) ∈ B . In other words, E ⊂ B
R j R j j Rn Rn
and it now follows that ⊗n B = M(E) ⊂ B by Proposition 1.9.
k=1 R Rn
LetBdenotethecollectionofopenboxes, B = (a ,b )×···×(a ,b ) = (cid:81)n (a ,b ).
1 1 n n j=1 j j
Every open subset U ⊂ Rn is equal to a countable union of open boxes (just take all
the open boxes contained in U having rational vertices). Hence B ⊂ M(B). If
Rn
B = (cid:81)n (a ,b ) is an open box and E = (a ,b ) (so that B = (cid:81)n E ), then, from
j=1 j j j j j j=1 j
the description of π−1(E ) in the first part of the proof,
j j
n
(cid:92)
B = π−1(E ) ∈ M(E).
j j
j=1
Hence
B ⊂ M(B) ⊂ M(E) = ⊗n B .
Rn j=1 R
(cid:3)
8 MAA6616 COURSE NOTES FALL 2015
2. Measures
Definition 2.1. LetX be a set andM aσ-algebra onX. AmeasureonM is a function
µ : M → [0,+∞] such that
(i) µ(∅) = 0,
(ii) If (E )∞ is a sequence of disjoints sets in M, then
j j=1
(cid:32) (cid:33)
∞ ∞
(cid:91) (cid:88)
µ E = µ(E ).
j j
j=1 j=1
If µ(X) < ∞, then µ is finite. If X = ∪∞ X with µ(X ) < ∞ for each j, then µ is
j=1 j j T
σ-finite.
Almost all of the measures of importance in analysis are σ-finite.
A triple (X,M,µ) where X is a set, M is a σ-algebra and µ a measure on M, is a
F
measure space. (cid:47)
Here are some simple measures and some procedures for producing new measures
A
from old. Non-trivial examples of measures will have to wait for the Caratheodory and
Hahn-Kolmogorov theorems in the following sections.
Example 2.2. (a) Let X be any set and, for E ⊂ X, let |E| denote the cardinality of
R
E, in the sense of a finite number or ∞. The function µ : 2X → [0,+∞] defined by
(cid:40)
|E| if E is finite
µ(E) =
D ∞ if E is infinite
is a measure on (X,2X), called counting measure. It is finite if and only if X is finite,
and σ-finite if and only if X is countable.
(b) Let X be an uncountable set and M the σ-algebra of (at most) countable and co-
countable sets (Example 1.8(b)). The function µ : M → [0,∞] defined by µ(E) = 0
if E is countable and µ(E) = +∞ is E is co-countable is a measure.
(c) Let (X,M,µ) be a measure space and E ∈ M. Recall M from Example 1.8(c).
E
The function µ (A) := µ(A∩E) is a measure on (E,M ). (Why is the assumption
E E
E ∈ M necessary?)
(d) (Linear combinations) If µ is a measure on M and c > 0, then (cµ)(E) =: cµ(E) is
a measure, and if µ ,...µ are measures on the same M, then
1 n
(µ +···µ )(E) := µ (E)+···µ (E)
1 n 1 n
is a measure. Likewise a countably infinite sum of measures (cid:80)∞ µ is a measure.
n=1 n
(The proof of this last fact requires a small amount of care. See Problem 7.8.)
(cid:52)
One can also define products and pull-backs of measures, compatible with the con-
structions of product and pull-back σ-algebras. These examples will be postponed until
we have built up some more machinery of measurable functions.
MAA6616 COURSE NOTES FALL 2015 9
Theorem 2.3 (Basic properties of measures). Let (X,M,µ) be a measure space.
(a) (Monotonicity) If E,F ∈ M and E ⊂ F, then µ(F) = µ(F \ E) + µ(E). In
particular, µ(E) ≤ µ(F) and if µ(E) < ∞, then µ(F \E) = µ(F)−µ(E).
(b) (Subadditivity) If (E )∞ ⊂ M, then µ((cid:83)∞ E ) ≤ (cid:80)∞ µ(E ).
j j=1 j=1 j j=1 j
(c) (Monotone convergence for sets) If (E )∞ ⊂ M and E ⊂ E ∀j, then limµ(E )
j j=1 j j+1 j
exists and moreover µ(∪E ) = limµ(E ).
j j
(d) (Dominated convergence for sets) If (E )∞ is a decreasing (E ⊃ E for all j)
j j=1 j j+1
from M and µ(E ) < ∞, then limµ(E ) exists and moreover µ(∩E ) = limµ(E ).
1 j j j
Proof. a) By additivity, µ(F) = µ(F \E)+µ(E) ≥ µ(E). T
b) For 1 ≤ j ≤ g, let
(cid:32) (cid:33)
j−1 F
(cid:91)
F = E \ E .
j j k
k=1
A
Byproposition1.7, theF arepairwisedisjoint, F ⊂ E forallj and∪∞ F == ∪∞ E .
j j j j=1 j j=1 j
Thus by countable additivity and (a),
(cid:32) ∞ (cid:33) (cid:32)R∞ (cid:33) ∞ ∞
(cid:91) (cid:91) (cid:88) (cid:88)
µ E = µ F = µ(F ) ≤ µ(E ).
j j j j
j=1 j=1 j=1 j=1
D
c) With the added assumption that the sequence (E )∞ is nested increasing, (cid:83)j F =
j j=1 k=1 k
E for each j. Thus, by countable additivity,
j
(cid:32) (cid:33) (cid:32) (cid:33)
∞ ∞
(cid:91) (cid:91)
µ E = µ F
j j
j=1 j=1
∞
(cid:88)
= µ(F )
k
k=1
j
(cid:88)
= lim µ(F )
k
j→∞
k=1
(cid:32) (cid:33)
j
(cid:91)
= lim µ F
k
j→∞
k=1
= lim µ(E ).
j
j→∞
d) The sequence µ(E ) is decreasing (by (a)) and bounded below, so limµ(E ) exists.
j j
Let F = E \ E . Then F ⊂ F for all j, and (cid:83)∞ F = E \ (cid:84)∞ E . So by (c)
j 1 j j j+1 j=1 j 1 j=1 j
10 MAA6616 COURSE NOTES FALL 2015
applied to the F , and since µ(E ) < ∞,
j 1
∞ ∞
(cid:92) (cid:92)
µ(E )−µ( E ) = µ(E \ E )
1 j 1 j
j=1 j=1
= limµ(F )
j
= lim(µ(E )−µ(E ))
1 j
= µ(E )−limµ(E ).
1 j
Again since µ(E ) < ∞, it can be subtracted from both sides. (cid:3)
1
T
Remark 2.4. Note that in item (d) of Theorem 2.3, the hypothesis “µ(E ) < ∞” can
1
be replaced by “µ(E ) < ∞ for some j”. However the finiteness hypothesis cannot be
j
removed entirely. For instance, consider (N,2N) equipped with counting measure, and
let E = {k : k ≥ j}. Then µ(E ) = ∞ for all j but µ(F(cid:84)∞ E ) = µ(∅) = 0. (cid:5)
j j j=1 j
For any set X and subset E ⊂ X, there is a function 1 : X → {0,1} defined by
E
(cid:40) A
1 if x ∈ E
1 (x) = ,
E
0 if x (cid:54)∈ E
called the characteristic function or Rindicator function of E. For a sequence of subsets
(E ) of X, by definition (E ) converges to E pointwise if 1 → 1 pointwise1. This
n n En E
notion allows the formulation of a more refined version of the dominated convergence
theorem for sets, which foreshadows (and is a special case of) the dominated convergence
D
theorem for the Lebesgue integral. See Problems 7.11 and 7.12.
Definition 2.5. Let (X,M,µ) be a measure space. A null set is a set E ∈ M with
µ(E) = 0. (cid:47)
It follows immediately from countable subadditivity that a countable union of null
sets is null. The contrapositive of this statement is a measure-theoretic version of the
pigeonhole principle:
Proposition 2.6 (Pigeonhole principle for measures). If (E )∞ is a sequence of sets
n n=1
in M and µ(∪E ) > 0, then µ(E ) > 0 for some n. †
n n
It will often be tempting to assert that if µ(E) = 0 and F ⊂ E, then µ(F) = 0, but
one must be careful: F need not be a measurable set. This caveat is not a big deal in
practice, however, because we can always enlarge the σ-algebra on which a measure is
defined so as to contain all subsets of null sets, and it will usually be convenient to do
so.
Definition 2.7. If(X,M,µ)hasthepropertythatF ∈ M wheneverE ∈ M,µ(E) = 0,
and F ⊂ E, then µ is complete. (cid:47)
1What would happen if we asked for uniform convergence?