8/2/2019 WEBdynkin

1/5

Tutorial 1: Dynkin systems 1

1. Dynkin systems

Definition 1 A Dynkin system on a set is a subset D of thepower set P(), with the following properties:

(i) D

(ii) A,B D, A B B \ A D

(iii) An D, An An+1, n 1

+n=1

An D

Definition 2 A -algebra on a set is a subset F of the powerset P() with the following properties:

(i) F

(ii) A F Ac= \ A F

(iii) An F, n 1 +

n=1

An F

www.probability.net

http://www.probability.net/http://www.probability.net/

8/2/2019 WEBdynkin

2/5

Tutorial 1: Dynkin systems 2

Exercise 1. Let F be a -algebra on . Show that F, thatif A,B F then A B F and also A B F. Recall that

B \ A = B Ac

and conclude that F is also a Dynkin system on .Exercise 2. Let (Di)iI be an arbitrary family of Dynkin systems

on , with I= . Show that D= iIDi is also a Dynkin system on

.

Exercise 3. Let (Fi)iI be an arbitrary family of-algebras on ,

with I= . Show that F= iIFi is also a -algebra on .

Exercise 4. Let A be a subset of the power set P(). Define:

D(A)

= {D Dynkin system on : A D}Show that P() is a Dynkin system on , and that D(A) is not empty.Define:

D(A)=

DD(A)

D

www.probability.net

http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/

8/2/2019 WEBdynkin

3/5

Tutorial 1: Dynkin systems 3

Show that D(A) is a Dynkin system on such that A D(A), andthat it is the smallest Dynkin system on with such property, (i.e.

if D is a Dynkin system on with A D, then D(A) D).

Definition 3 Let A P(). We call Dynkin system generatedbyA, the Dynkin system on, denotedD(A), equal to the intersectionof all Dynkin systems on , which contain A.

Exercise 5. Do exactly as before, but replacing Dynkin systems by-algebras.

Definition 4 Let A P(). We call -algebra generated byA, the -algebra on , denoted (A), equal to the intersection of all-algebras on , which contain A.

Definition 5 A subsetA of the power set P() is called a-systemon , if and only if it is closed under finite intersection, i.e. if it hasthe property:

A,B A A B A

www.probability.net

http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/

8/2/2019 WEBdynkin

4/5

Tutorial 1: Dynkin systems 4

Exercise 6. Let A be a -system on . For all A D(A), we define:

(A

)

= {B

D(A) :A

B

D(A)}1. IfA A, show that A (A)

2. Show that for all A D(A), (A) is a Dynkin system on .

3. Show that ifA A, then D(A) (A).4. Show that ifB D(A), then A (B).

5. Show that for all B D(A), D(A) (B).

6. Conclude that D(A) is also a -system on .

Exercise 7. Let D be a Dynkin system on which is also a -system.

1. Show that ifA,B D then A B D.

www.probability.net

http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/

8/2/2019 WEBdynkin

5/5

Tutorial 1: Dynkin systems 5

2. Let An D, n 1. Consider Bn= n

i=1Ai. Show that+n=1An =

+n=1Bn.

3. Show that D is a -algebra on .

Exercise 8. Let A be a -system on . Explain why D(A) is a

-algebra on , and (A) is a Dynkin system on . Conclude thatD(A) = (A). Prove the theorem:

Theorem 1 (Dynkin system) LetC be a collection of subsets of which is closed under pairwise intersection. If D is a Dynkin system

containing C then D also contains the -algebra (C) generated by C.

www.probability.net

http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/http://www.probability.net/