MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions )

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1 MATHEMATICS 117/E-217, SPRING 2012 PROBABILITY AND RANDOM PROCESSES WITH ECONOMIC APPLICATIONS Module #5 (Borel Field and Measurable Functions ) Last modified: May 1, 2012 Reading Dineen, pages Proof (may appear on an exam) Define the Borel sigma-field on R and prove that it is generated by the countable collection of open intervals {(, q) : q Q}. Warmups(intended to be done before lecture) Show that the subset of R that contains the single point π and the closed interval [3.5,3.7] belongs to the Borel sigma-field on [3,4]. (Just look at the definitions on page 58.) Let A be the event Dineen s pride, at odds 3:1, wins the Kentucky Derby and define X = 81 A. Thus X is the value of a $2 win ticket on Dineen s pride. Let B be the Borel set (-1,0.5). Convince yourself that X 1 (B) belongs to the sigma-field generated by A. (See page 61. This is trivial once you understand the notation.) You are playing a game where you roll a single die. The payoff function X is $1 if the roll is an odd number, $10 if the roll is an even number, plus an extra $5 if the roll is greater than 3. Define an equivalence relation by saying that x y if the rolls have the same payoff. What partition of Ω = {1, 2, 3, 4, 5, 6} is generated by this equivalence relation? (See page 64.) 1

2 Lecture topics. 1. Random Variables and Measurable Functions One of our important goals will be to calculate the expected value of a random variable. For example, the expected value of a single throw of a dart can be obtained by taking all possible scores on the dart board (values of a random variable), multiplying each score by the probability of its occurrence, and summing. Clearly this calculation would be impossible if an event like the dart hits in the region where the score is 20 is not in the sigma-field. More generally, we want any event like the dart hits where the score is between 5 and 30 to be in the sigma-field, and so our choice of sigma field places a restriction on the scoring functions. The ones that are consistent with our chosen sigma-field are called measurable. Looking at the situation the other way around, the requirement that the scoring function for the sample space that corresponds to throwing a dart at a dart board be measurable imposes a requirement on a sigma-field, namely that an event like the dart hits in the 10-point region must be in the sigma-field. This connection between sigma-fields and measurable functions is independent of any specific probability function. It is the last topic that we will consider before introducing a probability space. 2. Borel sets and the Borel sigma-field So far we have steered clear of uncountable sample spaces except to mention that 2 Ω is too big to be a useful sigma-field when Ω is uncountable. Now we address the case where Ω = R. This case is important for its own sake, e.g. where the experiment is watch this unstable nucleus and see when it decays, but it also important because R is the codomain for a measurable function. The definition of the Borel sigma-field on R is that it is generated by the open intervals. However, the special role for open sets that this definition implies is illusory, since The intersection of all open intervals of the form (a 1, b + 1 ) is the n n closed interval [a, b]. The intersection of all open intervals of the form (x 1, x + 1 ) is the n n set that contains only the one point x. The union of all closed intervals of the form [a + 1, b 1 ] is the open n n interval (a, b). 2

3 So any countable union of open intervals, closed intervals, and isolated points is a Borel set. What is hard, in fact, is constructing a set that is not a Borel set, but such sets exist! It is especially convenient to know (your proof) that the intervals of the form (, q), where q ranges over the countable set of rational numbers, generate the Borel sigma-field. 3. Inverse images and measurable functions. Following a near-universal convention, we denote measurable functions and random variables by capital letters from near the end of the alphabet; e.g. X : Ω R. A point in the domain (the sample space) is traditionally denoted by ω. It is not in general a point in any R n. If C is a subset of the codomain of function f, the inverse image of C is the subset of the domain A that is mapped into C: f 1 (C) = {x A : f(x) C} Careful: there is no implication that X is an invertible function. Frequently, f 1 (C) is much larger than a single point or even a single interval. For example, if f(x) = cos x, f 1 ([0.5, 1]) is a countably infinite union of intervals. Any X : Ω R generates a sigma-field F X whose members are the inverse images of Borel sets. As a simple example, if a dart board is divided into three zones worth 0, 5, and 10 points respectively, the scoring function X generates a sigma-field in which a typical event is the score is not zero (the inverse image of the Borel set {5, 10}.) If F X F we say that X is F-measurable. To test for measurability, do not try to use the entire Borel field. To show that X is not F measurable, it is sufficient to find any Borel set (for example, a single point) whose inverse image is not in F. To show that X is F measurable, find the simplest collection of Borel sets that generate the image of X. For example, if the image is a finite or countable set of points, show that the inverse image of each point is in F. If the image of X is uncountable, consider the inverse image of an arbitrary set of the form (, q). 3

4 4. Indicator functions The indicator function of an event A Ω, denoted 1 A, has the value 1 if ω A, 0 otherwise. The inverse image of {1} is A; the inverse image of {0} is A c. For an arbitrary Borel set, the inverse image simply depends on whether or not 0 and 1 are or are not included on the set. For example, the inverse image of {0.3, 0.7} is the empty set. Elementary set operations can be recast in the language of indicator functions, as follows: 1 A B = 1 A 1 B. 1 A B = 1 A + 1 B 1 A 1 B. 1 A c = 1 1 A. Quite generally, we can make new F-measurable functions from old by means of addition, subtraction and multiplication. As a special case, we can make F-measurable functions by taking finite linear combinations of indicator functions on events in F. If Ω is finite, such functions necessarily have finite image. The level sets of the form X 1 ({c}) form a partition of Ω, though perhaps it is a coarser partition than the partition P X that generates F X. This could happen, for example, if P X includes {1} and {2} individually, but X = 1 {1,2} so that X(1) = X(2) = 1 and the only level set is {1, 2}. Now we have a new way to create sigma-fields. Define a simple function, a finite linear combination of indicator functions X = n c i 1 Ai. i=1 The level sets of X partition Ω into a partition B j, and each set in the partition has an associated value d j. A convenient way to represent X is in the form X = n d j 1 Bj. j=1 4

5 5. Positive and negative functions It will be convenient later in the course to deal with functions that never have negative values. This can be achieved by writing X = X + X where X + = X when X is not negative, 0 otherwise, while X = X when X is not positive, 0 otherwise. With these definitions, X = X + X. X +, X, and X. If X is F-measurable, so are 6. Summary of results for countable partition or countable range (Dineen proposition 4.14) (a) If F is generated by a countable partition, then any F-measurable function X must be constant on each set A n in the partition and must therefore have countable range. (b) Conversely, if X has countable range, it can be expressed in the form X = x n 1 Bn n=1 when the sets B n are pairwise disjoint. They need not form a complete partition. If all the B n are are in sigma-field F, then X is F- measurable. (c) If X has countable range, then any function Y that is F X - measurable must be constant on the level sets of X. (d) One easy way to create a function Y that is constant on the level sets of X is to define Y (ω) = f(x(ω)), where f : R R is Borel measurable (the inverse image of a Borel set is a Borel set not a hard requirement to meet). In this case, for any Borel set B Y 1 (B) = (f(x)) 1 (B) = X 1 (f 1 (B)). As a really simple example, let f be the squaring function and let B = {4}. Then f 1 (B) = { 2, +2} and the level sets of X for 2 and +2 coalesce into a single level set for Y. 5

6 Sample problems. 1. Explain how to generate the interval (1, π] (a Borel set) from intervals of the form (, q) (with rational q). 6

7 2. Let Ω = {1,, 6}. Specify for me three subsets of Ω that are not pairwise disjoint and that do not generate the sigma-field 2 Ω. Dineen s example 4.10, which uses {1, 2}, {1, 4}, and {2, 3, 5}, meets these requirements. My task is to determine what sigma-field F is generated by your three sets. 3. Now specify a function X as a linear combination of indicator functions on your three subsets and I will write it as a linear combination of indicator functions for subsets that generate the sigma-field F. 7

8 Small group exercises. 1. The connection between measurable functions and sigma-fields. (a) Let Ω = {1,, 8}, and let X = 21 {1,3,5} +1 {2,3,5,7} 1 {3,7,8}. Determine the sigma-field F X. (b) Let Ω = {1,, 9}, and let F be the sigma-field corresponding to the partition {1, 2}, {3}, {4, 5, 8}, {6, 7, 9}. Invent a function X that is F- measurable, and write it as a linear combination of indicator functions. Then invent a function Y that is not F-measurable. (c) Let Ω = R and define X = 31 [0,6] 51 [4,8] + 21 [5,7]. Show that X is Borel-measurable and find a finite partition of R that generates F X. 8

9 2. Proofs for Borel set and measurable functions. (a) Consider the function X : Ω R. Show that the collection of inverse images of Borel subsets of R under X meets all the requirements for a sigma-field. (b) Consider X : Ω R, and let F be a sigma-field on Ω. Show that if A is a collection of subsets that generates the Borel field and A A, X 1 (A) F, then X is F-measurable. (c) Prove that if F is a sigma-field on Ω that is generated by a countable partition (A n ), then a real-valued function X on Ω is F-measurable if and only if X is constant on each A n. 9

10 Homework problems. 1. Dineen, exercise Dineen, exercise Let Ω = {1, 10} and let X = 61 {1,3,5} + 31 {1,5,7,9} 1 {2,4,7,9} Write X in the form X = n i=1 a i1 Ai where, if i j, a i a j and A i A j =. Find F x. 10

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