2. Caratheodory/'s Extension
Tutorial 2: Caratheodory’s Extension
1
2. Caratheodory’s Extension
In the following, Ω is a set. Whenever a union of sets is denoted
as
opposed to ∪, it indicates that the sets involved are pairwise disjoint.
Definition 6 A semi-ring on Ω is a subset S of the power set P(Ω)
with the following properties:
(i)
∅ ∈ S
(ii)
A, B ∈ S ⇒ A ∩ B ∈ S
n
(iii)
A, B ∈ S ⇒ ∃n ≥ 0, ∃Ai ∈ S : A \ B =
Ai
i=1
The last property (iii) says that whenever A, B ∈ S, there is n ≥ 0
and A1, . . . , An in S which are pairwise disjoint, such that A \ B =
A1 . . . An. If n = 0, it is understood that the corresponding union
is equal to ∅, (in which case A ⊆ B).
www.probability.net
Tutorial 2: Caratheodory’s Extension
2
Definition 7 A ring on Ω is a subset R of the power set P(Ω) with
the following properties:
(i)
∅ ∈ R
(ii)
A, B ∈ R ⇒ A ∪ B ∈ R
(iii)
A, B ∈ R ⇒ A \ B ∈ R
Exercise 1. Show that A ∩ B = A \ (A \ B) and therefore that a
ring is closed under pairwise intersection.
Exercise 2.Show that a ring on Ω is also a semi-ring on Ω.
Exercise 3.Suppose that a set Ω can be decomposed as Ω = A1
A2
A3 where A1, A2 and A3 are distinct from ∅ and Ω. Define
S1 = {∅, A1, A2, A3, Ω} and S2 = {∅, A1, A2 A3, Ω}. Show that S1
and S2 are semi-rings on Ω, but that S1 ∩ S2 fails to be a semi-ring
on Ω.
Exercise 4. Let (Ri)i∈I be an arbitrary family of rings on Ω, with
I = ∅. Show that R = ∩i∈IRi is also a ring on Ω.
www.probability.net
Tutorial 2: Caratheodory’s Extension
3
Exercise 5. Let A be a subset of the power set P(Ω). Define:
R(A) = {R ring on Ω : A ⊆ R}
Show that P(Ω) is a ring on Ω, and that R(A) is not empty. Define:
R(A) =
R
R∈R(A)
Show that R(A) is a ring on Ω such that A ⊆ R(A), and that it is
the smallest ring on Ω with such property, (i.e. if R is a ring on Ω
and A ⊆ R then R(A) ⊆ R).
Definition 8
Let A ⊆ P(Ω). We call ring generated by A, the
ring on Ω, denoted R(A), equal to the intersection of all rings on Ω,
which contain A.
Exercise 6.Let S be a semi-ring on Ω. Define the set R of all finite
unions of pairwise disjoint elements of S, i.e.
R = {A : A = ni=1Ai for some n ≥ 0, Ai ∈ S}
www.probability.net
Tutorial 2: Caratheodory’s Extension
4
(where if n = 0, the corresponding union is empty, i.e. ∅ ∈ R). Let
A = ni=1Ai and B = pj=1Bj ∈ R:
1. Show that A ∩ B =
i,j(Ai ∩ Bj) and that R is closed under
pairwise intersection.
2. Show that if p ≥ 1 then A \ B = ∩pj=1( ni=1(Ai \ Bj)).
3. Show that R is closed under pairwise difference.
4. Show that A ∪ B = (A \ B)
B and conclude that R is a ring
on Ω.
5. Show that R(S) = R.
Exercise 7. Everything being as before, define:
R = {A : A = ∪ni=1Ai for some n ≥ 0, Ai ∈ S}
www.probability.net
Tutorial 2: Caratheodory’s Extension
5
(We do not require the sets involved in the union to be pairwise dis-
joint). Using the fact that R is closed under finite union, show that
R ⊆ R, and conclude that R = R = R(S).
Definition 9 Let A ⊆ P(Ω) with ∅ ∈ A. We call measure on A,
any map μ : A → [0, +∞] with the following properties:
(i)
μ(∅) = 0
+∞
+∞
(ii)
A ∈ A, An ∈ A and A =
An ⇒ μ(A) =
μ(An)
n=1
n=1
The
indicates that we assume the An’s to be pairwise disjoint in
the l.h.s. of (ii). It is customary to say in view of condition (ii) that
a measure is countably additive.
Exercise 8.If A is a σ-algebra on Ω explain why property (ii) can
be replaced by:
+∞
+∞
(ii) An ∈ A and A =
An ⇒ μ(A) =
μ(An)
n=1
n=1
www.probability.net
Tutorial 2: Caratheodory’s Extension
6
Exercise 9. Let A ⊆ P(Ω) with ∅ ∈ A and μ : A → [0, +∞] be a
measure on A.
1. Show that if A1, . . . , An ∈ A are pairwise disjoint and the union
A = ni=1Ai lies in A, then μ(A) = μ(A1) + . . . + μ(An).
2. Show that if A, B ∈ A, A ⊆ B and B \A ∈ A then μ(A) ≤ μ(B).
Exercise 10. Let S be a semi-ring on Ω, and μ : S → [0, +∞] be a
measure on S. Suppose that there exists an extension of μ on R(S),
i.e. a measure ¯
μ : R(S) → [0, +∞] such that ¯μ|S = μ.
1. Let A be an element of R(S) with representation A = ni=1Ai
as a finite union of pairwise disjoint elements of S. Show that
n
¯
μ(A) =
i=1 μ(Ai)
2. Show that if ¯
μ : R(S) → [0, +∞] is another measure with
¯
μ|S = μ, i.e. another extension of μ on R(S), then ¯μ = ¯μ.
www.probability.net
Tutorial 2: Caratheodory’s Extension
7
Exercise 11. Let S be a semi-ring on Ω and μ : S → [0, +∞] be a
measure. Let A be an element of R(S) with two representations:
n
p
A =
Ai =
Bj
i=1
j=1
as a finite union of pairwise disjoint elements of S.
p
1. For i = 1, . . . , n, show that μ(Ai) =
j=1 μ(Ai ∩ Bj)
n
p
2. Show that
i=1 μ(Ai) =
j=1 μ(Bj )
3. Explain why we can define a map ¯
μ : R(S) → [0, +∞] as:
n
¯
μ(A) =
μ(Ai)
i=1
4. Show that ¯
μ(∅) = 0.
www.probability.net
Tutorial 2: Caratheodory’s Extension
8
Exercise 12. Everything being as before, suppose that (An)n≥1 is
a sequence of pairwise disjoint elements of R(S), each An having the
representation:
pn
An =
Akn , n ≥ 1
k=1
as a finite union of disjoint elements of S. Suppose moreover that
A = +∞
n=1An is an element of R(S) with representation A =
p
j=1Bj,
as a finite union of pairwise disjoint elements of S.
1. Show that for j = 1, . . . , p, Bj = ∪+∞
n=1 ∪pn
k=1 (Akn ∩ Bj) and
explain why Bj is of the form Bj = +∞
m=1Cm for some sequence
(Cm)m≥1 of pairwise disjoint elements of S.
+∞
p
2. Show that μ(B
n
j) =
n=1
k=1 μ(Akn ∩ Bj)
3. Show that for n ≥ 1 and k = 1, . . . , pn, Akn = pj=1(Akn ∩ Bj)
www.probability.net
Tutorial 2: Caratheodory’s Extension
9
p
4. Show that μ(Akn) =
j=1 μ(Akn ∩ Bj)
5. Recall the definition of ¯
μ of exercise (11) and show that it is a
measure on R(S).
Exercise 13.Prove the following theorem:
Theorem 2
Let S be a semi-ring on Ω. Let μ : S → [0, +∞] be a
measure on S. There exists a unique measure ¯
μ : R(S) → [0, +∞]
such that ¯
μ|S = μ.
www.probability.net
Tutorial 2: Caratheodory’s Extension
10
Definition 10
We define an outer-measure on Ω as being any
map μ∗ : P(Ω) → [0, +∞] with the following properties:
(i)
μ∗(∅) = 0
(ii)
A ⊆ B ⇒ μ∗(A) ≤ μ∗(B)
+∞
+∞
(iii)
μ∗
An ≤
μ∗(An)
n=1
n=1
Exercise 14. Show that μ∗(A ∪ B) ≤ μ∗(A) + μ∗(B), where μ∗ is
an outer-measure on Ω and A, B ⊆ Ω.
Definition 11 Let μ∗ be an outer-measure on Ω. We define:
Σ(μ∗) = {A ⊆ Ω : μ∗(T ) = μ∗(T ∩ A) + μ∗(T ∩ Ac) , ∀T ⊆ Ω}
We call Σ(μ∗) the σ-algebra associated with the outer-measure μ∗.
Note that the fact that Σ(μ∗) is indeed a σ-algebra on Ω, remains to
be proved. This will be your task in the following exercises.
www.probability.net
Tutorial 2: Caratheodory’s Extension
11
Exercise 15. Let μ∗ be an outer-measure on Ω. Let Σ = Σ(μ∗) be
the σ-algebra associated with μ∗. Let A, B ∈ Σ and T ⊆ Ω
1. Show that Ω ∈ Σ and Ac ∈ Σ.
2. Show that μ∗(T ∩ A) = μ∗(T ∩ A ∩ B) + μ∗(T ∩ A ∩ Bc)
3. Show that T ∩ Ac = T ∩ (A ∩ B)c ∩ Ac
4. Show that T ∩ A ∩ Bc = T ∩ (A ∩ B)c ∩ A
5. Show that μ∗(T ∩ Ac) + μ∗(T ∩ A ∩ Bc) = μ∗(T ∩ (A ∩ B)c)
6. Adding μ∗(T ∩(A∩B)) on both sides 5., conclude that A∩B ∈ Σ.
7. Show that A ∪ B and A \ B belong to Σ.
Exercise 16. Everything being as before, let An ∈ Σ, n ≥ 1. Define
B1 = A1 and Bn+1 = An+1 \ (A1 ∪ . . . ∪ An). Show that the Bn’s are
pairwise disjoint elements of Σ and that ∪+∞
n=1An =
+∞
n=1Bn.
www.probability.net
Tutorial 2: Caratheodory’s Extension
12
Exercise 17. Everything being as before, show that if B, C ∈ Σ and
B ∩ C = ∅, then μ∗(T ∩ (B C)) = μ∗(T ∩ B) + μ∗(T ∩ C) for any
T ⊆ Ω.
Exercise 18.Everything being as before, let (Bn)n≥1 be a sequence
of pairwise disjoint elements of Σ, and let B = +∞
n=1Bn. Let N ≥ 1.
1. Explain why
N
n=1Bn ∈ Σ
N
2. Show that μ∗(T ∩ ( N
n=1Bn)) =
n=1 μ∗(T ∩ Bn)
3. Show that μ∗(T ∩ Bc) ≤ μ∗(T ∩ ( N
n=1Bn)c)
+∞
4. Show that μ∗(T ∩ Bc) +
n=1 μ∗(T ∩ Bn) ≤ μ∗(T ), and:
+∞
5. μ∗(T ) ≤ μ∗(T ∩Bc)+μ∗(T ∩B) ≤ μ∗(T ∩Bc)+
n=1 μ∗(T ∩Bn)
+∞
6. Show that B ∈ Σ and μ∗(B) =
n=1 μ∗(Bn).
7. Show that Σ is a σ-algebra on Ω, and μ∗|Σ is a measure on Σ.
www.probability.net
Tutorial 2: Caratheodory’s Extension
13
Theorem 3
Let μ∗ : P(Ω) → [0, +∞] be an outer-measure on Ω.
Then Σ(μ∗), the so-called σ-algebra associated with μ∗, is indeed a
σ-algebra on Ω and μ∗|Σ(µ∗), is a measure on Σ(μ∗).
Exercise 19. Let R be a ring on Ω and μ : R → [0, +∞] be a
measure on R. For all T ⊆ Ω, define:
+∞
μ∗(T ) = inf
μ(An) , (An) is an R-cover of T
n=1
where an R-cover of T is defined as any sequence (An)n≥1 of elements
of R such that T ⊆ ∪+∞
n=1An. By convention inf ∅ = +∞.
1. Show that μ∗(∅) = 0.
2. Show that if A ⊆ B then μ∗(A) ≤ μ∗(B).
www.probability.net
Tutorial 2: Caratheodory’s Extension
14
3. Let (An)n≥1 be a sequence of subsets of Ω, with μ∗(An) < +∞
for all n ≥ 1. Given > 0, show that for all n ≥ 1, there exists
an R-cover (Apn)p≥1 of An such that:
+∞
μ(Apn) < μ∗(An) + /2n
p=1
Why is it important to assume μ∗(An) < +∞.
4. Show that there exists an R-cover (Rk) of ∪+∞
n=1An such that:
+∞
+∞ +∞
μ(Rk) =
μ(Apn)
k=1
n=1 p=1
+∞
5. Show that μ∗(∪+∞
n=1An) ≤ +
n=1 μ∗(An)
6. Show that μ∗ is an outer-measure on Ω.
www.probability.net
Tutorial 2: Caratheodory’s Extension
15
Exercise 20. Everything being as before, Let A ∈ R. Let (An)n≥1
be an R-cover of A and put B1 = A1 ∩ A, and:
Bn+1 = (An+1 ∩ A) \ ((A1 ∩ A) ∪ . . . ∪ (An ∩ A))
1. Show that μ∗(A) ≤ μ(A).
2. Show that (Bn)n≥1 is a sequence of pairwise disjoint elements
of R such that A = +∞
n=1Bn.
3. Show that μ(A) ≤ μ∗(A) and conclude that μ∗|R = μ.
Exercise 21. Everything being as before, Let A ∈ R and T ⊆ Ω.
1. Show that μ∗(T ) ≤ μ∗(T ∩ A) + μ∗(T ∩ Ac).
2. Let (Tn) be an R-cover of T . Show that (Tn ∩ A) and (Tn ∩ Ac)
are R-covers of T ∩ A and T ∩ Ac respectively.
3. Show that μ∗(T ∩ A) + μ∗(T ∩ Ac) ≤ μ∗(T ).
www.probability.net
Tutorial 2: Caratheodory’s Extension
16
4. Show that R ⊆ Σ(μ∗).
5. Conclude that σ(R) ⊆ Σ(μ∗).
Exercise 22.Prove the following theorem:
Theorem 4 (Caratheodory’s extension) Let R be a ring on Ω
and μ : R → [0, +∞] be a measure on R. There exists a measure
μ : σ(R) → [0, +∞] such that μ|R = μ.
Exercise 23. Let S be a semi-ring on Ω. Show that σ(R(S)) = σ(S).
Exercise 24.Prove the following theorem:
Theorem 5
Let S be a semi-ring on Ω and μ : S → [0, +∞] be a
measure on S. There exists a measure μ : σ(S) → [0, +∞] such that
μ|S = μ.
www.probability.net
Document Outline
