142 CHAPTER 6. MEASURES AND MEASURABLE FUNCTIONS

Then µ (F) = ∑∞i=1 µ (Fi+1 \Fi) + µ (F1). Now µ (Fi+1 \Fi) + µ (Fi) = µ (Fi+1). If any

µ (Fi) = ∞, there is nothing to prove. Assume then that these are all finite. Then

µ (Fi+1 \Fi) = µ (Fi+1)−µ (Fi) sinceµ (Fi+1 \Fi)+µ (Fi) = µ (Fi+1)

and so

µ (F) =∞

∑i=1

µ (Fi+1)−µ (Fi)+µ (F1)

= limn→∞

n

∑i=1

µ (Fi+1)−µ (Fi)+µ (F1) = limn→∞

µ (Fn+1)

3.) Next suppose µ (F1) < ∞ and {Fn} is a decreasing sequence. Then F1 \Fn is in-creasing to F1 \F and so by the first part,

µ (F1)−µ (F) = µ (F1 \F) = limn→∞

µ (F1 \Fn)

= limn→∞

(µ (F1)−µ (Fn)) = µ (F1)− limn→∞

µ (Fn)

so limn→∞ µ (Fn) = µ (F) . ■

6.3 Dynkin’s LemmaDynkin’s lemma is a very useful result. It is used quite a bit in books on probability. Firstnote that if K is any collection of subsets of Ω which contains /0 and Ω, one can take theintersection of all σ algebras which contain K , one such being P (Ω). This intersectionis also a σ algebra and is denoted as σ (K ) and is the smallest σ algebra containing K .

Definition 6.3.1 Let Ω be a set and let K be a collection of subsets of Ω. Then Kis called a π system if /0,Ω ∈K and whenever A,B ∈K , it follows A∩B ∈K . σ (K )will denote the smallest σ algebra which contains K . More precisely, the intersection ofall σ algebras which contain K .

The following is the fundamental lemma which shows these π systems are useful. Thisis due to Dynkin.

Lemma 6.3.2 Let K be a π system of subsets of Ω, a non empty set. Also let G be acollection of subsets of Ω which satisfies the following three properties.

1. K ⊆ G

2. If A ∈ G , then AC ∈ G

3. If {Ai}∞

i=1 is a sequence of disjoint sets from G then ∪∞i=1Ai ∈ G .

Then G ⊇ σ (K ) , where σ (K ) is the smallest σ algebra which contains K .

Proof: First note that if H ≡ {G : 1 - 3 all hold} then ∩H yields a collection of setswhich also satisfies 1 - 3. Therefore, I will assume in the argument that G is the smallestcollection satisfying 1 - 3. Let A ∈K and define

GA ≡ {B ∈ G : A∩B ∈ G } .