Chapter 7 Infinite product spaces

8 CHAPTER 7. INFINITE PRODUCT SPACES
(1) A, B 2 S implies A \ B 2 S .
(2) If A 2 S , then there exist Ai 2 S disjoint such that A c = S n i=1 Ai.
It is easy to verify that semialgebras on W automatically contain W and ∆.
Lemma 7.13 The following are semialgebras:
(1) S = {(a, b] : • apple a apple b apple •}
(2) S = {A1 ⇥···⇥An : Ai 2 Ai} where Ai are algebras.
(3) S = ⇥ Aa : Aa 2 Aa &#{a : Aa 6= Wa} < • provided Aa’s are algebras.
a2I
Proof. Let us focus on (2): Since all Ai are p-systems, S is closed under intersections.
To show the complementation property in the definition of semialgebra,
let D be a collection of all sets of the form B1 ⇥···⇥Bn, where Bi is either Ai or A c i ,
but such that A1 ⇥···⇥An is not included in D. Then
(A1 ⇥···⇥An) c = [
From here the complementation property follows by noting that D is finite and
D ⇢ S .
Next we will show that via finite disjoint unions semialgebras give rise to algebras:
Lemma 7.14 Let S be a semialgebra and let S be the set of finite unions of sets Ai 2 S .
Then S is an algebra.
Proof. First we show that S is closed under intersections. Indeed, A is the (finite)
union of some Ai 2 S and B is the (finite) union of some Bj 2 S . Therefore,
✓
[
A \ B =
i
Ai
◆
\
B2B
B.
✓ ◆
[
Bj =
j
[
Ai \ Bj.
i,j
Since Ai, Bj 2 S , then Ai \ Bj 2 S and A \ B is the finite union of sets from S .
Consequently, A \ B 2 S .
Next we will show that S is closed under taking the complement. Let A be as
above. Then
A c ✓
[
=
◆c = \ \ [
= Bi,j,
i
Ai
where Bi,j 2 S are such that A c i is the union of Bi,j—see the definition of semialgebra.
The union on the extreme right is finite and, therefore, it belongs to S . But
S is closed under finite intersections, so A c 2 S as well.
Now we will learn how (and under what conditions) a set function on a semialgebra
can be extended to a measure on the associated algebra:
i
A c i
i
j