221 Analysis 2, 2008–09
Summary of theorems and definitions
Measure theory
Definition. The extended real line is the set
[−∞, ∞] = R ∪ {−∞, ∞}
where −∞ and ∞ are two symbols which do not belong to R. We extend
order, addition and product on R to [−∞, ∞] in the natural way by
defining, for any x ∈ R:
(i). −∞ < x, x < ∞ and −∞ < ∞;
(ii). (−∞) + x = −∞ = x + (−∞), ∞ + x = ∞ = x + ∞, ∞ + ∞ = ∞
and (−∞) + (−∞) = −∞ (note that (−∞) + ∞ is not defined);


if x > 0
∞
(iii). ∞ · x = x · ∞ = 0
if x = 0 , ∞ · ∞ = ∞ = (−∞) · (−∞) and


−∞ if x < 0
(−∞) · y = −(∞ · y) = y · (−∞) for any y ∈ [−∞, ∞].
Definition. Let X be a set. We write P(X) = {all subsets of X}.
Definition. Let X be a set. A collection A ⊆ P(X) of subsets of X is
called a ring of subsets of X , or simply a ring , if
(i). ∅ ∈ A,
(ii). A, B ∈ A =⇒ A \ B ∈ A, and
(iii). A, B ∈ A =⇒ A ∪ B ∈ A.
Definition. A measure is a function m : A → [0, ∞], where A is a ring of
subsets of a set X, such that
(i). m(∅) = 0, and
∞
∞
[
X
(ii). m
Ai =
m(Ai )
i=1
• Ai ∈ A for i ≥ 1,
whenever
i=1
• Ai ∩ Aj = ∅ for i 6= j, and
S
• ∞
i=1 Ai ∈ A.
We abbreviate property (ii) by saying that m is
countably additive on disjoint unions , or just countably additive .
1
Proposition. Let m : A → [0, ∞] be a measure, where A is a ring of
subsets of X. Then
(i). if A, B ∈ A with A ⊆ B, then m(A) ≤ m(B);
(ii). if A, B ∈ A with A ⊆ B and m(B) < ∞ then
m(B \ A) = m(B) − m(A).
Lemma (The disjoint unions trick). If A is a ring of subsets of a set X
ei ∈ A with
and Ai ∈ A for i = 1, 2, . . . then there are pairwise disjoint sets A
ei ⊆ Ai
A
∞
[
and
ei =
A
i=1
∞
[
Ai .
i=1
Theorem (Measures are countably subadditive). Let m : A → [0, ∞] be a
measure where A is a ring. For any Ai ∈ A, i = 1, 2, . . . , we have
∞
∞
[
X
m
Ai ≤
m(Ai ).
i=1
i=1
Definition. If Ai ⊆ X for i = 1, 2, . . . and A ⊆ X,
Ai ↑ A means A1 ⊆ A2 ⊆ . . .
Ai ↓ A means A1 ⊇ A2 ⊇ . . .
and A =
and A =
∞
[
i=1
∞
\
Ai ;
Ai .
i=1
Theorem (Increasing unions and decreasing intersections).
Let m : A → [0, ∞] be a measure, where A is a ring of subsets of X, and let
Ai ∈ A for i = 1, 2, . . . .
(i). If Ai ↑ A where A ∈ A, then m(Ai ) → m(A) as i → ∞.
(ii). If Ai ↓ A where A ∈ A and if m(A1 ) < ∞, then m(Ai ) → m(A) as
i → ∞.
Definition. Let X be a set. A collection A ⊆ P(X) of subsets of X is
called a σ-algebra if
(i). ∅ ∈ A,
(ii). A ∈ A =⇒ A{ ∈ A [we write A{ = X \ A], and
S
(iii). Ai ∈ A for i = 1, 2, . . . =⇒ ∞
i=1 Ai ∈ A.
Proposition.
T
If A is a σ-algebra and Ai ∈ A for i = 1, 2, . . . then ∞
i=1 Ai ∈ A.
2
Definition. Let m : A → [0, ∞] be a measure where A is a ring of subsets
of a set X. The outer-measure of m is the function m∗ : P(X) → [0, ∞]
defined for any Y ⊆ X by
m∗ (Y ) = inf
∞
nX
m(Ai ) : Ai ∈ A for i = 1, 2, . . . and Y ⊆
i=1
∞
[
o
Ai .
i=1
Theorem. If A ∈ A then m∗ (A) = m(A). So m∗ extends m.
Theorem. (i). If A, B ⊆ X with A ⊆ B then m∗ (A) ≤ m∗ (B).
(ii). m∗ is countably subadditive. That is, if Yi ⊆ X for i = 1, 2, . . . then
m
∗
∞
[
Yi ≤
i=1
∞
X
m∗ (Yi ).
i=1
Definition. Let m : A → [0, ∞] be a measure where A is a ring of subsets
of a set X. A set B ⊆ X is m-measurable if for any Y ⊆ X,
m∗ (Y ) = m∗ (Y ∩ B) + m∗ (Y \ B).
no proof Theorem (The Extension Theorem). Let m : A → [0, ∞] be a measure
where A is a ring of subsets of a set X, and let M denote the collection of
all m-measurable subsets of X.
(i). A ⊆ M,
(ii). M is a σ-algebra, and
(iii). m∗ |M : M → [0, ∞] is a measure which extends m.
Corollary. For any measure m : A → [0, ∞], there is an extension of m to
a measure on the σ-algebra M of all m-measurable sets.
Definition. Let m : A → [0, ∞] be a measure where A is a ring of subsets
of a set X. We say that m is σ-finite if there are sets Xi ∈ A with Xi ↑ X
such that m(Xi ) < ∞ for i = 1, 2, . . . .
Theorem (Uniqueness of extensions for σ-finite measures).
For any σ-finite measure m : A → [0, ∞], there is a unique extension of m
to a measure on the σ-algebra M of all m-measurable sets.
3
Lebesgue measure on R
Definition. The interval ring is the set I ⊆ P(R),
I = {all finite unions of intervals of the form (a, b] where a, b ∈ R}.
Definition. The length measure m : I → [0, ∞] is defined by
m((a1 , b1 ] ∪ (a2 , b2 ] ∪ · · · ∪ (an , bn ]) = b1 − a1 + b2 − a2 + . . . + bn − an
where a1 < b1 < a2 < b2 < · · · < an < bn .
no proof Theorem. The length measure m : I → [0, ∞] is a (well-defined) σ-finite
measure.
Definition. Let m : I → [0, ∞] be the length measure.
A set B ⊆ R is Lebesgue measurable if it is m-measurable.
Let M denote the collection of all Lebesgue measurable sets.
Lebesgue measure m : M → [0, ∞] is the measure obtained by extending
the length measure to the σ-algebra M of Lebesgue measurable sets using
the Extension Theorem.
For the rest of this section, we will write m for Lebesgue measure and M
for the collection of all Lebesgue measurable sets.
Proposition. Let A ∈ M and c ∈ R.
(i). m(A + c) = m(A) [m is translation invariant], and
(ii). m(cA) = |c|m(A).
no proof Theorem. There is a subset of R which is not Lebesgue measurable.
Definition. Let S ⊆ P(R) be any collection of subsets of R.
The σ-algebra generated by S is
o
\n
σ(S) =
A ⊆ P(R) : A is a σ-algebra and S ⊆ A
Proposition. σ(S) is the smallest σ-algebra of subsets of R which
contains S. That is,
(i). σ(S) is a σ-algebra and S ⊆ σ(S), and
(ii). If B ⊆ P(R) is any σ-algebra with S ⊆ B then σ(S) ⊆ B.
Definition. The collection of Borel subsets of R is defined to be σ(O)
where O is the collection of open subsets of R.
Theorem. The collection of Borel subsets of R is equal to σ(I). So
I ⊆ σ(I) = σ(O) ⊆ M ( P(R).
Theorem. For any Lebesgue measurable set A ∈ M, there is a Borel set B
with A ⊆ B and m(B \ A) = 0.
4
Integration with respect to a measure
Definition. A measure space consists of a set X, a σ-algebra M ⊆ P(X)
and a measure m : M → [0, ∞].
Let us fix a measure space (X, M, m). We refer to the sets in M as
measurable sets.
Definition. A function f : X → [−∞, ∞] is measurable if the set
f −1 (α, ∞] = {x ∈ X : f (x) > α}
is in M for every α ∈ R.
Definition. If A ⊆ X then the characteristic function of A is the function
χA : X → R defined by
(
1
if x ∈ A,
χA (x) =
0
if x 6∈ A.
Proposition. If f, g : X → [−∞, ∞] are two measurable functions, A ∈ M
is a measurable set and c ∈ [−∞, ∞], then f + g, f χA and cf are all
measurable functions.
Definition. A function ϕ : X → R is simple if the range of ϕ is a finite
set.
Proposition. Let ϕ : X → R be a simple function. There is a partition
X = A1 ∪ · · · ∪ An of X and a1 , . . . , an ∈ R with ai 6= aj if i 6= j such that
ϕ = a1 χ1 + · · · + an χn .
(F)
Apart from re-ordering, there is only one way to write ϕ in this form.
Moreover, ϕ is measurable if and only if each set A1 , . . . , An is measurable.
Definition. If ϕ : X → [0, ∞) is a nonnegative, simple measurable function
written in the form (F) then the integral of ϕ with respect to m is
defined as
Z
ϕ dm = a1 m(A1 ) + · · · + an m(An ).
If A is a measurable set then ϕχA is a nonnegative, simple measurable
function, so we can define the integral of ϕ over A with respect to m by
Z
Z
ϕ dm =
ϕχA dm.
A
Proposition. If A is any measurable set then
5
R
χA dm = m(A).
Proposition. If c ∈ [0, ∞) and ϕ, ψ : X → [0, ∞) are nonnegative, simple
measurable functions then
(i). Rϕ + ψ is nonnegative,
simple
and measurable, and
R
R
ϕ + ψ dm = ϕ dm + ψ dm, and
R
R
(ii). cϕ is nonnegative, simple and measurable, and cϕ dm = c ϕ dm.
Theorem. If ϕ : X → [0, ∞) is a nonnegative, simple measurable function
then the map λ : M → [0, ∞],
Z
λ(A) =
ϕ dm, for A ∈ M
A
is a measure.
Corollary. Let ϕ : X → [0, ∞) be a nonnegative, simple measurable
function. Let A ∈ M and An ∈ M for n = 1, 2, . . . , with An ↑ A. Then
Z
Z
ϕ dm →
ϕ dm as n → ∞.
An
A
Definition. Let f : X → [0, ∞] be a nonnegative measurable function. The
integral of f with respect to m is defined as
Z
nZ
o
ϕ : X → [0, ∞) is a nonnegative, simple
f dm = sup
ϕ dm :
.
measurable function with 0 ≤ ϕ ≤ f
If A is a measurable set then f χA is a nonnegative measurable function, so
we can define the integral of f over A with respect to m by
Z
Z
f dm = f χA dm.
A
Proposition. Let f and g be nonnegative measurable functions X → [0, ∞]
and let A, B ∈ M.
R
R
(i). If f ≤ g then f dm ≤ g dm.
R
R
(ii). If A ⊆ B then A f dm ≤ B f dm.
Definition. A sequence (fn )n≥1 of functions fn : X → [−∞, ∞] is
monotone increasing if f1 (x) ≤ f2 (x) ≤ f3 (x) ≤ . . . for every x ∈ X.
If f : X → [−∞, ∞], we write f = lim fn or fn → f as n → ∞ if
n→∞
fn (x) → f (x) for every x ∈ X.
Proposition. Let (fn )n≥1 be a monotone increasing sequence of functions
fn : X → [−∞, ∞].
(i). There is a function f : X → [−∞, ∞] with f = lim fn .
n→∞
(ii). If each fn is a measurable function, then so is f = lim fn .
n→∞
6
Theorem (The Monotone Convergence Theorem). If (fn )n≥1 is a
monotone increasing sequence of nonnegative measurable functions
fn : X → [0, ∞] then
Z
Z
lim fn dm = lim
fn dm.
n→∞
n→∞
Theorem. Let f : X → [0, ∞] be any nonnegative measurable function.
There is a monotone increasing sequence (ϕn )n≥1 of nonnegative, simple
measurable functions ϕn : X → [0, ∞) with f = lim ϕn .
n→∞
Corollary. (i). If f, g : X → [0, ∞] are nonnegative measurable functions
and c ≥ 0 then f + g and cf are nonnegative measurable functions,
and
Z
Z
Z
Z
Z
f + g dm = f dm + g dm
and
cf dm = c f dm.
(ii). If fn :P
X → [0, ∞] are nonnegative measurable functions for n ≥ 1
then ∞
n=1 fn is a nonnegative measurable function and
Z X
∞
fn dm =
∞ Z
X
fn dm.
n=1
n=1
Corollary. If f : X → [0, ∞) is a nonnegative measurable function then the
map λ : M → [0, ∞],
Z
f dm, for A ∈ M
λ(A) =
A
is a measure.
Corollary. Let f : X → [0, ∞) be a nonnegative measurable function. Let
Ai ∈ M for i = 1, 2, . . . .
S
(i). If A = ∞
i=1 Ai is a countable disjoint union then
Z
f dm =
A
(ii). If Ai ↑ A then
R
f dm →
∞ Z
X
i=1
R
f dm.
Ai
f dm as i → ∞.
R
R
R
(iii). If Ai ↓ A and A1 f dm < ∞ then Ai f dm → A f dm as i → ∞.
Ai
A
7
Definition. A set A ∈ M is m-null , or null , if m(A) = 0.
Definition. Let f, g : X → [−∞, ∞] be measurable functions. We say that
f = g almost everywhere , or f = g a.e. if the set {x ∈ X : f (x) 6= g(x)}
is m-null.
Theorem. Let f : X → [0, ∞] be a nonnegative measurable function. Then
Z
f dm = 0 ⇐⇒ f = 0 almost everywhere.
Corollary. If A is an m-null
set and f : X → [0, ∞] is any nonnegative
R
measurable function then A f dm = 0.
Corollary. If f, g : X → [0, ∞]Rare nonnegative
measurable functions with
R
f = g almost everywhere, then f dm = g dm.
Theorem (An almost everywhere version of the MCT). If (fn )n≥1 is a
monotone increasing sequence of nonnegative measurable functions
fn : X → [0, ∞] with fn ↑ f almost everywhere then
Z
Z
f dm = lim
fn dm.
n→∞
Definition. If f : X → R is a measurable function, we say f is
integrable with respect to m if
Z
Z
+
f dm < ∞ and
f − dm < ∞
where f + , f − : X → [0, ∞) are the nonnegative measurable functions
(
(
f
(x)
if
f
(x)
≥
0
−f (x)
if f (x) ≤ 0
f + (x) =
and
f − (x) =
0
if f (x) < 0
0
if f (x) > 0
If f is integrable with respect to m, we define the
integral of f with respect to m by
Z
Z
Z
+
f dm = f dm − f − dm.
Proposition.
If fR1 , f2 : X → [0, ∞) are measurable functions with
R
f1 dm < ∞ and f2 dm < ∞ then the function f = f1 − f2 is integrable
and
Z
Z
Z
f dm = f1 dm − f2 dm.
Corollary. If f, g : X → R are integrable functions and c ∈ R then f + g
and cf are integrable with
Z
Z
Z
Z
Z
f + g dm = f dm + g dm
and
cf dm = c f dm.
8
Theorem. If f, gR : X → RRare integrable functions with f = g almost
everywhere then f dm = g dm.
Theorem. If f : X → R is measurable then |f | is measurable and
(i). f is integrable ⇐⇒ |f | is integrable; and
R
R
(ii). if f is integrable then | f dm| ≤ |f | dm.
Definition. A complex-valued function f : X → C is integrable if the
real-valued functions Re f and Im f are both integrable. If f is integrable,
the integral of f with respect to m is the complex number
Z
Z
Z
f dm = Re f dm + i Im f dm.
R
R
Theorem. If f : X → C is integrable then | f dm| ≤ |f | dm.
Definition. Let (an )n≥1 be a sequence in [−∞, ∞]. The lim sup and
lim inf of this sequence are defined by
lim sup an = lim sup ai
and
n→∞
n→∞
lim inf an = lim
n→∞
i≥n
n→∞
inf ai .
i≥n
Theorem (Fatou’s lemma). If (fn )n≥1 is a sequence of nonnegative
measurable functions fn : X → [0, ∞] then
Z
Z
lim inf fn dm ≤ lim inf fn dm.
n→∞
n→∞
Theorem (The Dominated Convergence Theorem). Suppose that (fn )n≥1 is
a sequence of integrable functions fn : X → R and that fn → f as n → ∞.
If there is an integrable function g : X → R with |fn | ≤ g for all n ≥ 1, then
Z
Z
f is integrable, and
f dm = lim
fn dm.
n→∞
Theorem.
functions
fn : X → R so
P∞ Let (fn )n≥1 be a sequence of integrableP
∞ R
that n=1 |fn (x)| < ∞ for every x ∈ X and with n=1 |fn | dm < ∞.
Then
Z X
∞ Z
∞
X
fn dm.
fn dm =
n=1
n=1
Theorem (Differentiation under the integral sign). Let f : X × [a, b] → R
be a function so that
(i). for each t ∈ [a, b], x 7→ f (x, t) is an integrable function X → R; and
(ii). for each x ∈ X, t 7→ f (x, t) is a differentiable function [a, b] → R; and
(iii). for some integrable g : X → R we have
∂f
(x, t) ≤ g(x)
∂t
for every t ∈ [a, b] and x ∈ X.
Then
d
dt
Z
Z
f (x, t) dx =
9
∂f
(x, t) dx
∂t
Lebesgue integration on R
Definition. The Lebesgue integral is the integral with respect to
Lebesgue measure m : M → R where M is the collection of Lebesgue
measurable subsets of R. In other words, integrals with respect to m are
defined by applying the preceding theory to the measure space (R, M, m)
where M and m are the Lebesgue measurable subsets of R, and Lebesgue
measure on R, respectively. For a ≤ b, we’ll write
Z
Z a
Z b
f (x) dx instead of
f dm
f (x) dx = −
b
a
(a,b)
Proposition. For any a, b, c ∈ R and any integrable function f : R → R,
Z c
Z c
Z b
f (x) dx +
f (x) dx =
f (x) dx.
a
b
a
Theorem (The Fundamental Theorem of
R t Calculus). Let a ∈ R, let
f : R → R be continuous and let F (t) = a f (x) dx for t ∈ R. Then
F 0 (t) = f (t) for all t ∈ R.
Corollary.
If G : R → R is continuously differentiable and a, b ∈ R then
Rb 0
G (x) dx = G(b) − G(a).
a
u
f
Corollary. If a, b, c, d ∈ R and u and f are functions [a, b] → [c, d] → R
with u continuously differentiable and f continuous, then
Z
b
Z
0
u(b)
f (u(x)) u (x) dx =
a
f (y) dy.
u(a)
Theorem. Let f : R → R be an integrable function and let c ∈ R.
R
R
(i). f (x + c) dx = f (x) dx
R
R
1
(ii). If c 6= 0 then f (cx) dx = |c|
f (x) dx.
10
Multiple integration
Let (X, L, `) and (Y, M, m) be measure spaces. We will write
Z
Z
Z
Z
f (x) d`(x) = f d` and
g(y) dm(y) = g dm
whenever the expressions on the right hand side make sense.
Definition. If A ∈ L and B ∈ M then the set A × B is called a
rectangle . We write rect(L, M) = {A × B : A ∈ L, B ∈ M} for the set of
all rectangles. We also write
Arect (L, M) = {all finite unions of rectangles A × B ∈ rect(L, M)}.
Proposition. Arect (L, M) is a ring of subsets of X × Y , and every element
of Arect (L, M) is a finite disjoint union of rectangles from rect(L, M).
S
Definition. If E ∈ Arect (L, M) with E = ∞
i=1 Ai × Bi is a disjoint union
(possibly with Ai × Bi = ∅ for i sufficiently large) then we define
X
π(E) =
`(Ai )m(Bi ).
i≥1
Theorem. π : Arect (L, M) → [0, ∞] is a well-defined measure.
Definition. The product σ-algebra of the σ-algebras L and M is
L × M = σ(Arect (L, M)).
The product measure of the measures ` and m is l × m = π ∗ |L×M , the
restriction of the outer measure π ∗ to L × M.
Since L × M is contained in the π-measurable subsets of X × Y , the
Extension Theorem shows that ` × m is a measure.
Monotone classes
Definition. Let X be a set. A monotone class of subsets of X is a
non-empty collection C ⊆ P(X) which is closed under increasing unions and
decreasing intersections:
(i). if An ∈ C and An ↑ A for some A ⊆ X, then A ∈ C; and
(ii). if An ∈ C then and An ↓ A for some A ⊆ X then A ∈ C.
If S ⊆ P(X) is a non-empty collection of subsets of X, we write mon(S)
for the smallest monotone class of subsets of X which contains S.
11
Proposition. (i). A non-empty collection of subsets of X which is closed
under increasing unions and complements is a monotone class.
(ii). Any σ-algebra is a monotone class.
(iii). S ⊆ mon(S) ⊆ σ(S) for any non-empty collection S of subsets of X.
no proof Theorem (Monotone class lemma). If A ⊆ P(X) is a ring of subsets of X
then mon(A) is also a ring of subsets of X.
Corollary. Let A be a ring of subsets of a set X. If X ∈ mon(A) then
mon(A) = σ(A).
Corollary. L × M = mon(Arect (L, M)).
Integration using product measure
Definition. Let E ⊆ X × Y . For x ∈ X and y ∈ Y , we write
Ex = {y ∈ Y : (x, y) ∈ E} and E y = {x ∈ X : (x, y) ∈ E}.
Theorem. If E ∈ L × M then Ex ∈ M for every x ∈ X, and E y ∈ L for
every y ∈ Y .
no proof Theorem. Let l and m be σ-finite measures and let E ∈ L × M.
(i). Both of the functions X → [0, ∞], x 7→ m(Ex ) and
Y → [0, ∞], y 7→ m(E y ) are measurable.
Z
Z
(ii). l × m(E) =
m(Ex ) dl(x) =
l(E y ) dm(y).
X
Y
Theorem (Tonelli’s theorem). If F : X × Y → [0, ∞] is a nonnegative
measurable function then
Z Z
Z
Z Z
F (x, y) dm(y) dl(x) =
F d(l×m) =
F (x, y) dl(y) dm(x).
X
Y
X×Y
Y
X
[In particular, all of the functions that must be measurable for these
integrals to be defined, are measurable!]
Theorem (Fubini’s theorem). If F : X × Y → R is integrable then
Z Z
Z
Z Z
F (x, y) dm(y) dl(x) =
F d(l×m) =
F (x, y) dl(y) dm(x).
X
Y
X×Y
Y
R
X
Here, the Rintegral Y F (x, y) dm(y) is defined for almost every x ∈ X,
and x 7→ Y F (x, y) dm(y)
R is integrable.
Similarly,R the integral X F (x, y) dl(x) is defined for almost every y ∈ Y ,
and y 7→ X F (x, y) dl(x) is integrable.
12
Lebesgue measure and integration on Rn
Definition. Let (R, M, m) be the usual Lebesgue measure space, as on
page 10. Lebesgue measure on R2 is the measure
m × m : M × M → [0, ∞]. We obtain a corresponding integral from the
measure space (R2 , M × M, m × m).
Similarly, we define Lebesgue measure on Rn by
m × · · · × m : M × · · · × M → [0, ∞], and obtain a corresponding integral
from the measure space (Rn , M × · · · × M, m × · · · × m).
If f : Rn → R is integrable then we write
Z
f (x) dx
Rn
for the integral of f with respect to Lebesgue measure on Rn .
We have the following generalisation of the final theorem on page 10:
Theorem. Let A be an invertible n × n matrix with real entries. If
f : Rn → R is integrable then
Z
Z
1
f (x) dx.
f (Ax) dx =
| det A| Rn
Rn
Corollary. Let A be any n × n matrix with real entries and let
µ = m × · · · × m denote Lebesgue measure on Rn . If S ⊆ Rn is a Lebesgue
measurable and we write A(S) = {Ax : x ∈ S} then
µ(A(S)) = | det A|µ(S).
13