Mathematics 721Fall 2010On Dynkin systemsDefinition. A Dynkin-system D on X is a collection of subsets of Xwhich has the following properties.(i) X ∈ D.(ii) If A ∈ D then its complement Ac:= X \ A belong to D.(iii) If Anis a s equ en ce of mutually disjoint sets in D then ∪∞n=1An∈ D.Observe that every σ-algebra is a Dynkin system.A1. In the literature one can also find a definition with alternative axioms(i), (ii)*, (iii)* where again (i) X ∈ D, and(ii)* I f A, B are in D and A ⊂ B then B \ A ∈ D.(iii)* If An∈ D, An⊂ An+1for all n = 1, 2, 3, . . . then also ∪∞n=1An∈ D.Prove that (i), (ii), (iii) is equivalent with (i), (ii)*, (iii)*.A2. Verify: If E is any collection of subsets of X then the intersectionof all Dynkin-systems containing E is a Dynkin system containing E. Itis the smallest Dynkin system containing E. We call it the Dynkin-systemgenerated by E, and denote it by D(E).Definition: A collection A of subsets of X is ∩-stable if for A ∈ A andB ∈ A we also have A ∩ B ∈ A.Observe that a ∩-stable system is stable under finite intersections.A3. (i) Show that if D is a ∩-stable Dynkin system, then the union oftwo sets in D is again in D.(ii) Prove: A Dynkin-system is a σ-algebra if and only if it is ∩-stable.The following theorem turns out to be very useful for the construction ofσ-algebras.A4. Theorem: Let E be any collection of subsets of X which is stableunder intersections. Then the Dynkin-system D(E) generated by E is equalto the σ-algebra M(E) g enerated by E.For the proof follow the following steps.(i) Argue that it suffices to show that D(E) is a σ-algebra. By ProblemA3 it suffices to show that D(E) is stable under intersections(ii) Fix a set B ∈ D(E). Prove that the systemΓB= {A ⊂ X : A ∩ B ∈ D(E)}is a Dynkin system.(iii) Prove that E ⊂ ΓBfor all B ∈ E, and hence D(E) ⊂ ΓBfor all B ∈ E.(iv) Prove that E ⊂ ΓBeven for all B ∈ D(E), and hence D(E) ⊂ ΓBforall B ∈ D(E). Conclude.12First assignment: Due September 17.A: Work through the above “project” on Dynkin systems and do problemsA1-4.B: Problems fr om chapter 1 in Folland: 3, 5, 8, 12, 14, 15, 16, 18, 19.Note th at in problem 3a “disjoint sets” should read “n onempty disjoint sets”.C1. Definition: A system R of subs ets of X is a ring (on X) if (a) ∅ ∈ R,(b) A, B ∈ R =⇒ A \ B ∈ R, and (c) A, B ∈ R =⇒ A ∪ B ∈ R,(i) Prove that a ring is closed und er finite intersections (i.e. A, B ∈ R =⇒A ∩ B ∈ R).(ii) Prove that a ring on X is an algebra if and only if X ∈ R.(iii) Consider the operation of symmetric difference and intersection(E, F ) 7→ E△ F := E \ F ∪ F \ E(E, F ) 7→ E ∩ F.Check the following facts about the symmetric difference:A△B = B△A(A△B)△C = A△(B△C)A△A = ∅; A△∅ = A(A△B) ∩ C = (A ∩ C)△(B ∩ C)Ac△Bc= A△B.Show th at if we define △ as an addition on a class R of subsets of X and∩ as a multiplication on R then R is a r ing on X if and only if (R, △, ∩) isa commutative ring in the algebraic sense. It is a ring with un it if and onlyif R is an algebra.Remark: This ring is a Boolean ring (every A ∈ R is idempotent: A△A =∅). If R has a unit can it happen that R is a field in the algebraic sense?(iv) A subclass N of a ring R on X is called an ideal in R if it satisfies(a) ∅ ∈ N ,(b) N ∈ N , M ∈ R, M ⊂ N =⇒ M ∈ N , and(c) M, N ∈ N =⇒ M ∪ N ∈ N .Show that N is an ideal in R if and only if N is an ideal in the commu-tative ring R in the algebraic sense.(v) Folland problem 1 c,d on page24.Additional example/exercise (not assigned): Let E = {A, B} where A andB are subsets of X.(i) Let R(E) be the smallest ring containing E (the ring generated by E- verify that this makes sense for general E). When is R(E) equal to theσ-algebra generated by E?(ii) Determine the Dynkin-system generated by E. Show that D(E) coin-cides with the σ-algebra generated by E if and on ly if one of the sets A ∩ B,A ∩ Bc, Ac∩ B, Ac∩ Bcis
View Full Document
Unlocking...