Mathematics 721 Fall 2016

On Dynkin systems Definition. A Dynkin-system D on X is a collection of subsets of X which has the following properties. (i) X ∈ D. (ii) If A ∈ D then its complement Ac := X \ A belong to D. ∞ (iii) If An is a sequence 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)* If A, B are in D and A ⊂ B then B \ A ∈ D. ∞ (iii)* If An ∈ D, An ⊂ An+1 for all n = 1, 2, 3,... then also ∪n=1An ∈ D. Prove that the definition with (i), (ii), (iii) is equivalent with the definition with (i), (ii)*, (iii)*.

A2. Verify: If E is any collection of subsets of X then the intersection of all Dynkin-systems containing E is a Dynkin system containing E. It is the smallest Dynkin system containing E. We call it the Dynkin-system generated by E, and denote it by D(E).

Definition: A collection A of subsets of X is ∩-stable if for A ∈ A and B ∈ 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 of two 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 of σ-algebras (we will use it later in the proof of Fubini’s theorem). A4. Theorem: Let E be any collection of subsets of X which is stable under intersections. Then the Dynkin-system D(E) generated by E is equal to the σ-algebra M(E) generated by E. For the proof work out the following steps. (i) Argue that it suffices to show that D(E) is a σ-algebra. By Problem A3 it suffices to show that D(E) is ∩-stable. (ii) Fix a set B ∈ D(E). Prove that the system

ΓB = {A ⊂ X : A ∩ B ∈ D(E)} is a Dynkin system. (iii) Prove that E ⊂ ΓB for all B ∈ E, and hence D(E) ⊂ ΓB for all B ∈ E. (iv) Prove that E ⊂ ΓB even for all B ∈ D(E), and hence D(E) ⊂ ΓB for all B ∈ D(E). Conclude. 1 2

First homework assignment: Work through the above “project” on Dynkin systems and do problems A1-4.

Additional problems: (not graded) B: Problems from chapter 1 in Folland: 3, 5, 8, 12, 14, 15, 16, 18, 19. Note that in problem 3a “disjoint sets” should read “nonempty disjoint sets”. C1. Definition: A system R of subsets 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 under 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→ E4F := E \ F ∪ F \ E (E,F ) 7→ E ∩ F. Check the following facts about the symmetric difference: A4B = B4A (A4B)4C = A4(B4C) A4A = ∅; A4∅ = A (A4B) ∩ C = (A ∩ C)4(B ∩ C) Ac4Bc = A4B. Show that if we define 4 as an addition on a class R of subsets of X and ∩ as a multiplication on R then R is a ring on X if and only if (R, 4, ∩) is a commutative ring in the algebraic sense. It is a ring with unit if and only if R is an algebra. Remark: This ring is a Boolean ring (every A ∈ R is idempotent: A4A = ∅). 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.

D. Let E = {A, B} where A and B 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 the σ-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 only if one of the sets A ∩ B, A ∩ Bc, Ac ∩ B, Ac ∩ Bc is nonempty.