Math 549, Measure theory and integration.
Basics of measure theory. Rigorous definitions.
ˆ Σ is said to be a σ -algebra of a set Ω iff Σ is a nonempty collection of subsets of Ω
that satisfy the following two properties:
(i) A ∈ Σ ⇒ Ac = Ω \ A ∈ Σ
S∞
(ii) {Ai }∞
i=1 ⊂ Σ ⇒
i=1 Ai ∈ Σ
Note that (i) and (ii) imply the property
T∞
(iii) {Ai }∞
i=1 ⊂ Σ ⇒
i=1 Ai ∈ Σ
∞
c c
Indeed, ∩∞
∈ Σ.
i=1 Ai = ∪i=1 (Ai )
ˆ A σ-algebra B of a metric space Ω is said to be Borel if it is generated by all open
subsets of Ω, that is, B is the smallest σ-algebra that contains all open subsets of Ω
(the intersection of all σ-algebras that contain the open subsets of Ω).
ˆ (Ω, Σ) is called a measurable space.
ˆ Let (Ω, Σ) and (R, B) be measurable spaces, where B is the Borel σ-algebra.
A function f : Ω → R is called a measurable map (or a random variable) iff
f −1 (A) ∈ Σ, ∀A ∈ B.
If, in addition, Ω = R and Σ = B, then f is called Borel.
Properties of random variables:
a sum of random variables is a random variable;
a product of random variables is a random variable;
a Borel function of a random variable is a random variable;
sup, inf, lim sup, lim inf of sequences of random variables are random variables.
ˆ A function µ : Σ → [0, ∞] is said to be a measure iff
(i) µ(∅) = 0
(ii) {Ai }∞
i=1 ⊂ Σ, Ai ∩ Aj = ∅ for i 6= j ⇒ µ
S∞
P∞
A
= i=1 µ(Ai )
i
i=1
Note: in other areas, a set function µ : Σ → C ∪ {∞} satisfying (i) and (ii) is also
called a measure.
ˆ (Ω, Σ, µ) is called a measure space.
If µ(Ω) = 1, then µ is called a probability measure, Ω is called a sample space, and Σ
is called an event space.
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ˆ Let (X1 , Σ1 , µ1 ) and (X2 , Σ2 , µ2 ) be measure spaces. Let F be the σ-algebra generated
by the rectangles A × B, where A ∈ Σ1 , B ∈ Σ2 . Such F is called the product σalgebra of Σ1 , Σ2 and denoted F = Σ1 × Σ2 . There is a unique measure µ on F
satisfying µ(A × B) = µ1 (A)µ2 (B) for all A ∈ Σ1 , B ∈ Σ2 . Such measure µ is called
the product measure and denoted µ = µ1 × µ2 .
Basics of integration.
Summary of the Riemann integral
Let R = R[a, b] denote the space of Riemann integrable functions on [a, b].
ˆ Basic properties:
Z b
dx = b − a (length of interval)
(1)
a
Z b
f (x) dx ≥ 0
(2) f ∈ R[a, b], f ≥ 0 ⇒
a
Z b+h
Z b
(3) f ∈ R[a, b] ⇒
f (x − h) dx ∀h ∈ R
f (x) dx =
a+h
a
Z b
Z b
(4) f ∈ R[a, b], c ∈ R ⇒ cf ∈ R,
a
Z b
Z b
(5) f, g ∈ R ⇒ f + g ∈ R,
f (x) dx
cf (x) dx = c
a
Z b
g(x) dx
f (x) dx +
(f (x) + g(x)) dx =
a
a
a
ˆ Which functions are Riemann integrable?
C[a, b] ⊂ R[a, b].
A bounded function is Riemann integrable if and only if the Lebesgue measure of
the set where it is discontinuous has measure 0.
Ex.: The indicator function of rational numbers 1Q ∈
/ R[a, b].
ˆ Changing the order of integral and limit:
{fn }∞
n=1 ⊂ R[a, b], fn ⇒ f (uniformly) on [a, b] =⇒ f ∈ R[a, b] and
Z b
Z b
Z b
lim
fn (x) dx =
lim fn (x) dx =
f (x) dx.
n→∞
a
a n→∞
a
Why do we need the Lebesgue integral?
satisfies (1)–(5);
enlarges the collection of integrable functions (in particular, includes R[a, b] ∪ {1Q });
enlarges the collection of sets for which length/area/volume can be calculated;
relaxes assumptions in convergence results (very useful technical feature).
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Informal comparison of the Riemann and Lebesgue integral constructions.
Riemann: partition domain and deal with changes of f relative to small changes of x (suggests
continuity).
Lebesgue: partition range and deal with changes of f arising from changes of x for which
the “length” can be measured (suggests measurability).
Summary of the Lebesgue integral (see sections 1.4–1.7 in [Durrett])
Let (Ω, Σ, µ) be a measure space and let L(Ω, µ) denote the space of Lebesgue integrable
functions on Ω with respect to µ.
If S is a finite or infinite interval in R and µ isZthe Lebesgue measure on S (i.e., µ(a, b) = b−a),
then the Lebesgue integral of f is denoted
f (x) dx.
S
Z
Z
The integral of f ∈ L(Ω, µ) is denoted
f (x) dµ(x) =
f dµ.
S
S
(If you are unfamiliar with the Lebesgue integral, you can think of the Riemann–Stieltjes
integral or of the Riemman integral with dµ(x) = g(x) dx.)
ˆ Which functions are Lebesgue integrable?
With respect to the Lebesgue measure:
(i) f ∈ R[a, b] ⇒ f ∈ L(Ω, dx) and the Riemann and Lebesgue integrals coincide.
(ii) If f : [0, ∞) → R belongs to R[a, b] for all 0 < a < b < ∞ and if the improper
Riemann integral of |f | exists and is finite, then f is Lebesgue integrable on [0, ∞)
and the respective Lebesgue and improper Riemann integrals of |f | coincide.
With respect to an abstract measure:
Pk
n
n
(i)
i=1 ai χAi ∈ L(S, µ), where k ∈ N, {ai }i=1 ⊂ R, {Ai }i=1 ⊂ Σ.
(ii) Every bounded measurable function is Lebesgue integrable over a set of finite
measure.
ˆ Change of variables formula:
Let (Ω, Σ) be a measurable space, X : Ω → R a random variable with distribution µ,
f : R → R a measurable function satisfying E |f (X)| < ∞. Then,
Z
E f (X) =
f (y)dµ(y).
R
ˆ Fubini’s theorem:
Let (X, Σ1 , µ1 ) and (Y, Σ2 , µ2 ) be measure spaces and µ =Z µ1 × µ2 . Let f : X × Y → R
be measurable (with respect to Σ1 × Σ2 ). If f ≥ 0 or
|f | dµ < ∞, then
X×Y
Z Z
Z
Z Z
f (x, y) dµ2 (y) dµ1 (x) =
X
Y
f dµ =
X×Y
3
f (x, y) dµ1 (x) dµ2 (y).
Y
X
ˆ Useful formulas:
Let (Ω, Σ, µ) be a measure
space, A,Z B ∈ Σ, µ(A
Z
Z ∩ B) = 0,Zf : Ω → R
Z integrable with
respect to µ. Then,
f 1A dµ =
Ω
Z
If µ(A) = 0, then
f dµ = 0.
f dµ,
A
f dµ =
A∪B
f dµ +
A
f dµ.
B
A
ˆ L p -norm of a random variable X on (Ω, Σ, P) with distribution µ is
Z
1/p Z
1/p
p 1/p
p
p
kXkp := (E |X| ) =
|X(ω)| dP(ω)
=
|x| dµ(x)
,
Ω
R
kXk∞ = ess sup |X| := inf M > 0 : |X| ≤ M a.e. .
p ∈ [1, ∞),
Here “a.e.” stands for almost everywhere, which means that a property holds on a
set of full measure (in our case, on a set of measure 1). In probability we also use a
synonymous term almost surely which is abbreviated “a.s.”.
If X is bounded everywhere, then kXk∞ = sup |X(ω)|.
ω∈Ω
ˆ L p -space Lp (Ω, Σ, P) consists of all measurable random variables X such that kXkp <
∞. The Lp -space equipped with the norm k · kp is a Banach space when p ∈ [1, ∞].
The L2 -space is a Hilbert space with the inner product hX, Y i := E XY .
Triangle (Minkowski’s) inequality for the Lp -norm:
kX + Y kp ≤ kXkp + kY kp ,
p ∈ [1, ∞].
Hölder’s inequality:
E |XY | ≤ kXkp kY kq ,
p, q ∈ [1, ∞],
1 1
+ = 1.
p q
When p = q = 2, we obtain the Cauchy-Schwarz inequality.
ˆ Recall that a function ϕ : R → R is said to be convex iff
ϕ(λx + (1 − λ)y) ≤ λϕ(x) + (1 − λ)ϕ(y) ∀λ ∈ [0, 1], ∀x, y ∈ R.
Jensen’s inequality:
Let X be a random variable and ϕ : R → R a convex (⇒ measurable) function such
that X and ϕ(X) are integrable. Then,
ϕ(E (X)) ≤ E (ϕ(X)).
In particular, if X is integrable, then
|E X| ≤ E |X|.
(Property of a Lebesgue integral:
a measurable function X is integrable if and only if |X| is integrable.)
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