1Some measure theory
III Advanced Probability
1.1 Review of measure theory
To make the course as self-contained as possible, we shall begin with some review
of measure theory. On the other hand, if one doesn’t already know measure
theory, they are recommended to learn the measure theory properly before
starting this course.
Definition
(
σ
-algebra)
.
Let
E
be a set. A subset
E
of the power set
P
(
E
) is
called a σ-algebra (or σ-field) if
(i) ∅ ∈ E;
(ii) If A ∈ E, then A
C
= E \ A ∈ E;
(iii) If A
1
, A
2
, . . . ∈ E, then
S
∞
n=1
A
n
∈ E.
Definition (Measurable space). A measurable space is a set with a σ-algebra.
Definition
(Borel
σ
-algebra)
.
Let
E
be a topological space with topology
T
.
Then the Borel
σ
-algebra
B
(
E
) on
E
is the
σ
-algebra generated by
T
, i.e. the
smallest σ-algebra containing T .
We are often going to look at B(R), and we will just write B for it.
Definition (Measure). A function µ : E → [0, ∞] is a measure if
– µ(∅) = 0
– If A
1
, A
2
, . . . ∈ E are disjoint, then
µ
∞
[
i=1
A
i
!
=
∞
X
i=1
µ(A
i
).
Definition
(Measure space)
.
A measure space is a measurable space with a
measure.
Definition
(Measurable function)
.
Let (
E
1
, E
1
) and (
E
2
, E
2
) be measurable
spaces. Then
f
:
E
1
→ E
2
is said to be measurable if
A ∈ E
2
implies
f
−1
(
A
)
∈ E
1
.
This is similar to the definition of a continuous function.
Notation.
For (
E, E
) a measurable space, we write
mE
for the set of measurable
functions E → R.
We write
mE
+
to be the positive measurable functions, which are allowed to
take value ∞.
Note that we do not allow taking the values ±∞ in the first case.
Theorem.
Let (
E, E, µ
) be a measure space. Then there exists a unique function
˜µ : mE
+
→ [0, ∞] satisfying
– ˜µ(1
A
) = µ(A), where 1
A
is the indicator function of A.
– Linearity: ˜µ(αf + βg) = α˜µ(f) + β ˜µ(g) if α, β ∈ R
≥0
and f, g ∈ mE
+
.
–
Monotone convergence: iff
f
1
, f
2
, . . . ∈ mE
+
are such that
f
n
% f ∈ mE
+
pointwise a.e. as n → ∞, then
lim
n→∞
˜µ(f
n
) = ˜µ(f ).
We call
˜µ
the integral with respect to
µ
, and we will write it as
µ
from now on.
Definition
(Simple function)
.
A function
f
is simple if there exists
α
n
∈ R
≥0
and A
n
∈ E for 1 ≤ n ≤ k such that
f =
k
X
n=1
α
n
1
A
n
.
From the first two properties of the measure, we see that
µ(f) =
k
X
n=1
α
n
µ(A
n
).
One convenient observation is that a function is simple iff it takes on only finitely
many values. We then see that if f ∈ mE
+
, then
f
n
= 2
−n
b2
n
fc ∧ n
is a sequence of simple functions approximating
f
from below. Thus, given
monotone convergence, this shows that
µ(f) = lim µ(f
n
),
and this proves the uniqueness part of the theorem.
Recall that
Definition (Almost everywhere). We say f = g almost everywhere if
µ({x ∈ E : f(x) 6= g(x)}) = 0.
We say f is a version of g.
Example.
Let
`
n
=
1
[n,n+1]
. Then
µ
(
`
n
) = 1 for all 1, but also
f
n
→
0 and
µ(0) = 0. So the “monotone” part of monotone convergence is important.
So if the sequence is not monotone, then the measure does not preserve limits,
but it turns out we still have an inequality.
Lemma (Fatou’s lemma). Let f
i
∈ mE
+
. Then
µ
lim inf
n
f
n
≤ lim inf
n
µ(f
n
).
Proof. Apply monotone convergence to the sequence inf
m≥n
f
m
Of course, it would be useful to extend integration to functions that are not
necessarily positive.
Definition
(Integrable function)
.
We say a function
f ∈ mE
is integrable if
µ(|f|) ≤ ∞. We write L
1
(E) (or just L
1
) for the space of integrable functions.
We extend µ to L
1
by
µ(f) = µ(f
+
) − µ(f
−
),
where f
±
= (±f) ∧ 0.
If we want to be explicit about the measure and the
σ
-algebra, we can write
L
1
(E, Eµ).
Theorem
(Dominated convergence theorem)
.
If
f
i
∈ mE
and
f
i
→ f
a.e., such
that there exists g ∈ L
1
such that |f
i
| ≤ g a.e. Then
µ(f) = lim µ(f
n
).
Proof. Apply Fatou’s lemma to g − f
n
and g + f
n
.
Definition
(Product
σ
-algebra)
.
Let (
E
1
, E
1
) and (
E
2
, E
2
) be measure spaces.
Then the product
σ
-algebra
E
1
⊗E
2
is the smallest
σ
-algebra on
E
1
×E
2
containing
all sets of the form A
1
× A
2
, where A
i
∈ E
i
.
Theorem.
If (
E
1
, E
1
, µ
1
) and (
E
2
, E
2
, µ
2
) are
σ
-finite measure spaces, then there
exists a unique measure µ on E
1
⊗ E
2
) satisfying
µ(A
1
× A
2
) = µ
1
(A
1
)µ
2
(A
2
)
for all A
i
∈ E
i
.
This is called the product measure.
Theorem
(Fubini’s/Tonelli’s theorem)
.
If
f
=
f
(
x
1
, x
2
)
∈ mE
+
with
E
=
E
1
⊗ E
2
, then the functions
x
1
7→
Z
f(x
1
, x
2
)dµ
2
(x
2
) ∈ mE
+
1
x
2
7→
Z
f(x
1
, x
2
)dµ
1
(x
1
) ∈ mE
+
2
and
Z
E
f du =
Z
E
1
Z
E
2
f(x
1
, x
2
) dµ
2
(x
2
)
dµ
1
(x
1
)
=
Z
E
2
Z
E
1
f(x
1
, x
2
) dµ
1
(x
1
)
dµ
2
(x
2
)