3 Product Spaces Example 1.

3 Product Spaces

Example 1. A stochastic model for coin tossing. For a single trial,

Ω ={0,1}, A=P(Ω), ∀ω ∈Ω :P({ω}) = 1/2. (1) Forn ‘independent’ trials, (1) serves as a building-block,

Ω_i ={0,1}, A_i =P(Ω_i), ∀ω_i ∈Ω_i :P_i({ω_i}) = 1/2, and we define

Ω =

n

Y

i=1

Ω_i, A=P(Ω), ∀A ∈A:P(A) = |A|

|Ω|. Then

P(A₁× · · · ×A_n) = P₁(A₁)· · · · ·P_n(A_n) for all A_i ∈A_i.

Question: How to model an infinite sequence of trials? To this end, Ω =

∞

Y

i=1

Ω_i.

How to choose aσ-algebra Ain Ω and a probability measureP on (Ω,A)? A reason- able requirement is

∀n ∈N ∀A_i ∈A_i :

P(A₁× · · · ×A_n×Ω_n+1×Ω_n+2. . .) =P₁(A₁)· · · · ·P_n(A_n). (2) Unfortunately,

A=P(Ω)

is too large, since there exists no probability measure on (Ω,P(Ω)) such that (2) holds.

The latter fact follows from a theorem by Banach and Kuratowski, which relies on the continuum hypothesis, see Dudley (2002, p. 526). On the other hand,

A={A₁ × · · · ×A_n×Ω_n+1×Ω_n+2· · ·:n ∈N, A_i ∈A_i for i= 1, . . . , n} (3) is not aσ-algebra.

Given: a non-empty setI and measurable spaces (Ω_i,A_i) for i∈I. Put Y =[

i∈I

Ωi

and define

Y

i∈I

Ω_i ={ω ∈Y^I :ω(i)∈Ω_i for i∈I}.

Notation: ω = (ω_i)_i∈I for ω∈Q

i∈IΩ_i. Moreover, let

P0(I) ={J ⊂I :J non-empty, finite}.

The following definition is motivated by (3).

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Definition 1.

(i) Measurable rectangle

A =Y

j∈J

Aj × Y

i∈I\J

Ωi

with J ∈ P₀(I) and A_j ∈ A_j for j ∈ J. Notation: R class of measurable rectangles.

(ii) Product (measurable) space (Ω,A) with components (Ω_i,A_i),i∈I, Ω =Y

i∈I

Ω_i, A=σ(R).

Notation: A=N

i∈IAi, product σ-algebra.

Remark 1. The class Ris a semi-algebra, but not an algebra in general. See ¨Ubung 2.3.

Example 2. Obviously, (2) only makes sense if A contains the product σ-algebra Nn

i=1A_i. We will show that there exists a uniquely determined probability measure P on the product space (Q∞

i=1{0,1},N∞

i=1P({0,1})) that satisfies (2), see Remark 4.3.(ii). The corresponding probability space yields a stochastic model for the simple case of gambling, which was mentioned in the introductory Example I.2.

We study several classes of mappings or subsets that generate the productσ-algebra.

Moreover, we characterize measurability of mappings that take values in a product space.

Put Ω =Q

i∈IΩi. For any ∅ 6=S ⊂I let π^I_S : Ω→Y

i∈S

Ω_i, (ω_i)i∈I 7→(ω_i)i∈S

denote the projection of Ω onto Q

i∈SΩ_i (restriction of mappings ω). In particular, for i ∈ I the i-th projection is given by π_{i}^I . Sometimes we simply write π_S instead of π^I_S and πi instead ofπ{i}.

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Theorem 1.

(i) N

i∈IA_i =σ({π_i :i∈I}).

(ii) ∀i∈I :A_i =σ(E_i) ⇒ N

i∈IA_i =σ S

i∈Iπ_i⁻¹(E_i) .

Proof. Ad (i), ‘⊃’: We show that every projection π_i : Ω → Ω_i is N

i∈IA_i -A_i- measurable. For A_i ∈A_i

π_i⁻¹(A_i) =A_i× Y

i∈I\{i}

Ω_i ∈R.

Ad (i), ‘⊂’: We show that R ⊂ σ({π_i : i ∈ I}). For J ∈ P₀(I) and A_j ∈ A_j with j ∈J

Y

j∈J

A_j× Y

i∈I\J

Ω_i =\

j∈J

π⁻¹_j (A_j).

Ad (ii): By Lemma 2.1 and (i) O

i∈I

A_i =σ[

i∈I

π_i⁻¹(A_i)

=σ[

i∈I

σ(π_i⁻¹(E_i))

=σ[

i∈I

π_i⁻¹(E_i) .

Corollary 1.

(i) For every measurable space (eΩ,A) and every mappinge g :Ωe →Ω g is A-e O

i∈I

A_i-measurable ⇔ ∀i∈I :π_i◦g is A-Ae _i-measurable.

(ii) For every ∅ 6=S⊂I the projection π_S^I is N

i∈IA_i-N

i∈SA_i-measurable.

Proof. Ad (i): Follows immediately from Theorem 2.3 and Theorem 1.(i).

Ad (ii): Note that π_{i}^S ◦π^I_S =π_i^I and use (i).

Remark 2. From Theorem 1.(i) and Corollary 1 we get O

i∈I

Ai =σ({π_S^I :S ∈P0(I)}).

The sets

π^I_S⁻¹

(B) =B × Y

i∈I\S

Ω_i withS ∈P₀(I) andB ∈N

i∈SA_iare calledcylinder sets. Notation: Cclass of cylinder sets. The classC is an algebra in Ω, but not aσ-algebra in general. Moreover,

R⊂α(R)⊂C⊂σ(R), where equality does not hold in general.

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Every product measurable set is countably determined in the following sense.

Theorem 2. For everyA∈ ⊗i∈IA_i there exists a non-empty countable setS ⊂I and a set B ∈ ⊗i∈SA_i such that

A= π_S^I−1 (B).

Proof. Put Ae =n

A∈O

i∈I

A_i :∃S ⊂I non-empty, countable ∃B ∈O

i∈S

A_i :A= π_S^I−1

(B)o .

By definition,Ae contains every cylinder set andAe ⊂N

i∈IA_i. It remains to show that Ae is a σ-algebra. Obviously, Ω∈A, and ife A = (π_S^I)⁻¹(B),A^c = (π_S^I)⁻¹(B^c). Finally, if An = (π^I_Sn)⁻¹(Bn), we define S =S

nSn and Ben = (π_S^S_n)⁻¹(Bn) =Bn×Q

i∈S\Bn ∈ N

i∈SA_i (see Corollary 1, (ii)); then

\

n

A_n=\

n

(π_S^I)⁻¹(Be_n) = ((π^I)_S)⁻¹ \

n

Be_n

! ,

hence T

nAn ∈A.e

Now we study products of Borel-σ-algebras.

Theorem 3.

B_k =

k

O

i=1

B, B_k=

k

O

i=1

B.

Proof. ByRemarkrefch2s1.refch2r5, B_k=σnY^k

i=1

]−∞, a_i] :a_i ∈R for i= 1, . . . , ko

⊂

k

O

i=1

B.

On the other hand, πi : R^k → R is continuous, hence it remains to apply Corollary 2.1 and Theorem 1.(i). Analogously,B_k =Nk

i=1Bfollows.

Remark 3. Consider a measurable space (Ω,e A) and a mappinge f = (f₁, . . . , f_k) :Ωe →R^k.

Then, according to Theorem 3, f is A-Be _k-measurable iff all functions f_i are A-B-e measurable.

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