Measure Theory (I): σ-algebras
11 Sep 2018
The central concept of measure theory is that of a measure on a set Ω, which is a way to quantify the “size” of its subsets. However, it is usually impossible to assign such a “size” to every subset of Ω in a way consistent with the axioms of a measure. In this post, we define the notion of a σ-algebra, which we think of as the collection of all “measurable” subsets of Ω.
Preliminaries
Let Ω be a set. The power set of Ω, denoted P(Ω), is the collection of all subsets of Ω.
A σ-algebra (or sigma-algebra) on Ω is a set Σ⊆P(Ω), satisfying:
- ∅∈Σ;
- A∈Σ⟹Ω\A∈Σ;
- A1,A2,…∈Σ⟹⋃k=1∞Ak∈Σ.
In other words, a σ-algebra is a collection of subsets of Ω containing ∅, and is closed under complement and countable union. Note that closure under countable union also implies closure under finite union, by taking Ak=∅ for all k>N.
Here are some examples of σ-algebras.
- Clearly, P(Ω) is a σ-algebra on Ω.
- By the axioms, any σ-algebra on Ω must contain ∅ and Ω\∅=Ω. Conversely, we can check that {∅,Ω} is a σ-algebra on Ω.
- Let A⊆Ω. Then {∅,A,Ω\A,Ω} is a σ-algebra on Ω.
Example: Let ∣Ω∣=4, and let
Σ={A⊆Ω:∣A∣=0,2,or 4}.
Then Σ contains ∅, and is closed under complement and countable disjoint union. However, the union of two 2-element subsets of Ω can contain 3 elements, so Σ is not a σ-algebra on Ω.
Now we prove some elementary consequences of the axioms.
Proposition: For any σ-algebra Σ on a set Ω, we have:
- A,B∈Σ⟹A\B∈Σ;
- A1,A2,…∈Σ⟹⋂k=1∞Ak∈Σ.
Proof: Note that
A\Bk=1⋂∞Ak=A∩(Ω\B),=Ω\k=1⋃∞(Ω\Ak). □
In other words, σ-algebras are closed under countable intersection, and thus finite intersection (by taking Ak=Ω for k>N).
Proposition: If {Σα}α∈I are σ-algebras on Ω, then so is ⋂α∈IΣα.
Proof: Check that each of the axioms of a σ-algebra carry over to the intersection. □
For any collection S⊆P(Ω) of subsets of Ω, we define the σ-algebra generated by S, denoted by σ(S), to be the intersection of all σ-algebras on Ω containing S.
Proposition: For any S⊆P(Ω), σ(S) is the smallest σ-algebra on Ω containing S.
Proof: By the previous proposition, σ(S) is a σ-algebra. By definition, any σ-algebra containing S must contain σ(S) as a subset. □
This gives a new way of creating σ-algebras:
-
Let S={{ω}:ω∈Ω} be the collection of singleton subsets of Ω. Then any σ-algebra containing S must contain all countable subsets of Ω, and their complements. Conversely, the collection
Σ={A⊆Ω:A or Ω\A countable}
can be checked to be a σ-algebra on Ω, so σ(S)=Σ.
- More generally, let S={Aα:α∈I} be a collection of disjoint subsets of Ω with union Ω. Then by a similar argument, σ(S) consists of all countable unions of Aα and their complements.
- Let Ω=Rn, and let On be the collection of open sets in Rn. Then B=Bn=σ(On) contains:
- all open and closed sets;
- all Gδ sets, ie. countable intersections of open sets;
- all Fσ sets, ie. countable unions of closed sets;
- all Gδσ sets, ie. countable unions of Gδ sets;
- all Fσδ sets, ie. countable intersections of Fσ sets;
- …
The elements of B are called Borel sets.
-
Let a=(a1,…,an), b=(b1,…,bn), where −∞≤ak≤bk≤∞ for each k=1,…,n. Define the half-open interval
[a,b):={(x1,…,xn)∈Rn:xk∈[ak,bk) for all k=1,…,n}.
Let Hn be the collection of all half-open intervals.
Proposition: Hn generates the σ-algebra of Borel sets, ie. σ(Hn)=Bn.
Proof: We give the proof for n=1; the case n>1 proceeds similarly.
First, note that
H1⊆O1⟹σ(H1)⊆σ(O1)=B1.
Now σ(H1) contains
k=1⋃∞[a+k1,b)=(a,b).
Note that the topology on R is generated by a countable basis of open intervals. Hence every open set in R is a countable union of open intervals, which lies in σ(H1). Thus
B1=σ(O1)⊆σ(H1),
so σ(H1)=B1. □
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