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 2 D. (ii) If A 2 D then its complement Ac := X n A belong to D. 1 (iii) If An is a sequence of mutually disjoint sets in D then [n=1An 2 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 2 D, and (ii)* If A; B are in D and A ⊂ B then B n A 2 D. 1 (iii)* If An 2 D, An ⊂ An+1 for all n = 1; 2; 3;::: then also [n=1An 2 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 2 A and B 2 A we also have A \ B 2 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 2 D(E). Prove that the system ΓB = fA ⊂ X : A \ B 2 D(E)g is a Dynkin system. (iii) Prove that E ⊂ ΓB for all B 2 E, and hence D(E) ⊂ ΓB for all B 2 E. (iv) Prove that E ⊂ ΓB even for all B 2 D(E), and hence D(E) ⊂ ΓB for all B 2 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) ; 2 R, (b) A; B 2 R =) A n B 2 R, and (c) A; B 2 R =) A [ B 2 R, (i) Prove that a ring is closed under finite intersections (i.e. A; B 2 R =) A \ B 2 R). (ii) Prove that a ring on X is an algebra if and only if X 2 R. (iii) Consider the operation of symmetric difference and intersection (E; F ) 7! E4F := E n F [ F n 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 2 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) ; 2 N , (b) N 2 N ;M 2 R;M ⊂ N =) M 2 N , and (c) M; N 2 N =) M [ N 2 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 = fA; Bg 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..