Lecture 1:Measurable spaces 1 of 13 Course: Theory of Probability I Term: Fall 2013 Instructor: Gordan Zitkovic Lecture 1 Measurable spaces Families of Sets 1 1 Definition 1.1 (Order properties). A countable family fAngn2N of To make sure we are all on the same subsets of a non-empty set S is said to be page, let us fix some notation and termi- nology: • ⊆ denotes a subset (not necessarily 1. increasing if An ⊆ An+1 for all n 2 N, proper). 2. decreasing if An ⊇ An+1 for all n 2 N, • A set A is said to be countable if there exists an injection (one-to-one mapping) from A into N. Note that 3. pairwise disjoint if An \ Am = Æ for m 6= n, finite sets are also countable. Sets 4.a partition of S if fA g is pairwise disjoint and [ A = S. which are not countable are called n n2N n n uncountable. • For two functions f : B ! C, g : A ! Here is a list of some properties that a family S of subsets of a B, the composition f ◦ g : A ! C of nonempty set S can have: f and g is given by ( f ◦ g)(x) = f (g(x)), (A1) Æ 2 S, for all x 2 A. (A2) S 2 S, • fAngn2N denotes a sequence. More generally, (Ag)g2G denotes a collec- (A3) A 2 S ) Ac 2 S, tion indexed by the set G. • We use the notation An % A to de- (A4) A, B 2 S ) A [ B 2 S, note that the sequence fAngn2N is increasing and A = [n An. Similarly, (A5) A, B 2 S, A ⊆ B ) B n A 2 S, An & A means that fAngn2N is de- creasing and A = \n An. (A6) A, B 2 S ) A \ B 2 S, (A7) An 2 S for all n 2 N )[n An 2 S, (A8) An 2 S, for all n 2 N and An % A implies A 2 S, (A9) An 2 S, for all n 2 N and fAngn2N is pairwise disjoint implies [n An 2 S, Definition 1.2 (Families of sets). A family S of subsets of a non-empty set S is called an 1. algebra if it satisfies (A1),(A3) and (A4), Last Updated: September 19, 2013 Lecture 1:Measurable spaces 2 of 13 2. s-algebra if it satisfies (A1), (A3) and (A7) 3. p-system if it satisfies (A6), 4. l-system if it satisfies (A2), (A5) and (A8). Problem 1.1. Show that: 1. Every s-algebra is an algebra. 2. Each algebra is a p-system and each s-algebra is an algebra and a l-system. 3. A family S is a s-algebra if and only if it satisfies (A1), (A3), (A6) and (A9). 4.A l-system which is a p-system is also a s-algebra. 5. There are p-systems which are not algebras. 6. There are algebras which are not s-algebras Hint: Pick all finite subsets of an infinite set. That is not an algebra yet, but sets 7. There are l-systems which are not p-systems. can be added to it so as to become an algebra which is not a s-algebra. Definition 1.3 (Generated s-algebras). For a family A of subsets of a non-empty set S, the intersection of all s-algebras on S that contain A is denoted by s(A) and is called the s-algebra generated 2 by A. 2 Since the family 2S of all subsets of S is a s-algebra, the concept of a gener- ated s-algebra is well defined: there is Problem 1.2. Show, by means of an example, that the union of a family always at least one s-algebra contain- of algebras (on the same S) does not need to be an algebra. Repeat for ing A - namely 2S. s(A) is itself a s- algebra (why?) and it is the smallest (in s-algebras, p-systems and l-systems. the sense of set inclusion) s-algebra that contains A. In the same vein, one can Definition 1.4 (Topology). A topology3 on a set S is a family t of define the algebra, the p-system and the l-system generated by A. The only im- subsets of S which contains Æ and S and is closed under finite inter- portant property is that intersections of sections and arbitrary (countable or uncountable!) unions. The ele- s-algebras, p-systems and l-systems are themselves s-algebras, p-systems and l- ments of t are often called the open sets. A set S on which a topology systems. is chosen (i.e., a pair (S, t) of a set and a topology on it) is called a 3 Almost all topologies in these notes topological space. will be generated by a metric, i.e., a set A ⊂ S will be open if and only if for each x 2 A there exists # > 0 such that Definition 1.5 (Borel s-algebras). If (S, t) is a topological space, then fy 2 S : d(x, y) < #g ⊆ A. The prime the s-algebra s(t), generated by all open sets, is called the Borel s- example is R where a set is declared open if it can be represented as a union 4 algebra on (S, t) or, less precisely , the Borel s-algebra on S. of open intervals. 4 We often abuse terminology and call S Example 1.6. Some important s-algebras. Let S be a non-empty set: itself a topological space, if the topology t on it is clear from the context. In the 1. The set S = 2S (also denoted by P(S)) consisting of all subsets of S same vein, we often speak of the Borel s-algebra on a set S. is a s-algebra. 2. At the other extreme, the family S = fÆ, Sg is the smallest s-algebra on S. It is called the trivial s-algebra on S. Last Updated: September 19, 2013 Lecture 1:Measurable spaces 3 of 13 3. The set S of all subsets of S which are either countable or whose complements are countable is a s-algebra. It is called the countable- cocountable s-algebra and is the smallest s-algebra on S which contains all singletons, i.e., for which fxg 2 S for all x 2 S. 4. The Borel s-algebra on R (generated by all open sets as defined by the Euclidean metric on R), is denoted by B(R). Problem 1.3. Show that the B(R) = s(A), for any of the following choices of the family A: 1. A = fall open subsets of Rg, Hint: An arbitrary open interval I = (a, b) in R can be written as I = A = f Rg −1 −1 2. all closed subsets of , [n2N[a + n , b − n ]. 3. A = fall open intervals in Rg, 4. A = fall closed intervals in Rg, 5. A = fall left-closed right-open intervals in Rg, 6. A = fall left-open right-closed intervals in Rg, 7. A = fall open intervals in R with rational end-pointsg, and 8. A = fall intervals of the form (−¥, r], where r is rationalg. Measurable mappings Definition 1.7 (Measurable spaces). A pair (S, S) consisting of a non- empty set S and a s-algebra S of its subsets is called a measurable space5. 5 If (S, S) is a measurable space, and A 2 S, we often say that A is measurable in S. Definition 1.8 (Pull-backs and push-forwards). For a function f : S ! T and subsets A ⊆ S, B ⊆ T, we define the 1. push-forward f (A) of A ⊆ S as f (A) = f f (x) : x 2 Ag ⊆ T, − 2. pull-back6 f 1(B) of B ⊆ T as 6 It is often the case that the notation is abused and the pull-back of B under f is f −1(B) = fx 2 S : f (x) 2 Bg ⊆ S. denoted simply by f f 2 Bg. This nota- tion presupposes, however, that the do- main of f is clear from the context. Problem 1.4. Show that the pull-back operation preserves the elemen- tary set operations, i.e., for f : S ! T, and B, fBngn2N ⊆ T, − − 1. f 1(T) = S, f 1(Æ) = Æ, Note: The assumption that the families in 2. and 3. above are countable is not nec- −1 −1 2. f ([nBn) = [n f (Bn), essary. Uncountable unions or intersec- −1 −1 tions commute with the pull-back, too. 3. f (\nBn) = \n f (Bn), and 4. f −1(Bc) = [ f −1(B)]c. Last Updated: September 19, 2013 Lecture 1:Measurable spaces 4 of 13 Give examples showing that the push-forward analogues of the state- ments 1., 3. and 4. above are not true. Definition 1.9 (Measurability). A mapping f : S ! T, where (S, S) and (T, T ) are measurable spaces, is said to be (S, T )-measurable7 if 7 When T = R, we tacitly assume that f −1(B) 2 S for each B 2 T . the Borel s-algebra is defined on T, and we simply call f measurable. In particu- lar, a function f : R ! R, which is mea- Proposition 1.10 (A measurability criterion). Let (S, S) and (T, T ) be surable with respect to the pair of the C T T = (C) Borel s-algebras is often called a Borel two measurable spaces, and let be a subset of such that s .