PROBABILITY THEORY THE HOMEWORK ASSIGNMENTS.

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1 PROBABILITY THEORY THE HOMEWORK ASSIGNMENTS. In all problems, please give complete and detailed solutions. Theorems proved in class may be used. Please, however, clearly formulate all theorems that you are using. THE SEVENTH ASSIGNMENT. Gnedenko, Problems: page , no. 1-7, p , no This homework is due on Tuesday 31 March in class. THE SIXTH ASSIGNMENT. Feller, Problems: p.130, no. 6, 7, 11, 12; p.159, no. 9, 10, 11, 16, 17; p.161, no. 30, 31, 32, 33, 34. Extra credit: p. 160, no. 21, 22. This homework is due on Tuesday 24 March in class.

2 THE FIFTH ASSIGNMENT (11 PROBLEMS). ALGEBRAS OF SETS. We start with a set X. Recall that a family C of subsets of X is called a ring if for any A, B C, we have A B C and A\B C; an algebra if, additionally, X C; and a sigma-algebra if, additionally, for any A 1,..., A n, C, we have A n C. 1. Let X, Y be two sets, let C be a sigma-algebra of subsets of Y. Show that the family f 1 C = {f 1 A A C} is also a sigma-algebra. 2. Let X be a set, and let A be an algebra of sets. Show that A is a sigmaalgebra if and only if one of the following equivalent conditions holds: 1) for any A n A such that A 1 A 2... we have A n A. 2) for any A n A such that A 1 A 2... we have A n A. 3. Let X be a set, let A be a sigma-algebra of sets, and let µ be a finitely additive measure on A such that µ(x) = 1. Show that µ is countably additive if and only if one of the following equivalent conditions holds: 1) for any A n A such that A 1 A 2... we have 2)for any A n A such that A 1 A 2... we have 4. A sigma-algebra is said to be countably generated (or separable) if it is generated by countably many sets. Show that the Borel sigma-algebra is countably generated. Remark. Recall that a sigma-algebra is said to be generated by a collection of sets if it is the smallest (by inclusion) sigma-algebra containing all those sets.

3 5. Let X be a set, let A be a sigma-algebra of sets, and let µ be a finitely additive measure on A such that µ(x) = 1. Show that µ is countably additive if and only if one of the following equivalent conditions holds: 1) for any A n A such that A 1 A 2... we have 2)for any A n A such that A 1 A 2... we have Remark. Don t forget to show that the limits at the right-hand side exist! 6. Give an example of a sigma-algebra containing exactly eight sets. MEASURABLE FUNCTIONS. Let X be a set, let A be a sigma-algebra of subsets of X, and let f : X R be a function. the function f is called measurable if the full pre-image of any Borel subset of R lies in the sigma-algebra A on X. Remark. Recall that the Borel sigma-algebra is the smallest sigmaalgebra containing all intervals in R. 7. Let f be measurable. Show that so is f. Is the converse true? 8. Let f n be bounded measurable functions. Show that so are supf n, inf f n. n n 9. Prove that the following conditions (assumed to hold for any a R) are equivalent to the measurability of f. 1) f 1 (, a) A; 2) f 1 (, a] A; 3) f 1 [a, ) A; 4) f 1 (a, ) A. 10. Let f : (0, 1) R be everywhere differentiable. Prove that its derivative is Borel measurable (i.e., is measurable with respect to the Borel sigmaalgebra on (0, 1)). 11. Let f : [a, b] [c, d] be continuous. for t [c, d], let n(t) be the number of solutions to the equation f(x) = t (and we set n(t) = 0 if that number is infinite). Prove that the function n : [c, d] R is measurable (with respect to the Borel sigma-algebra on [c, d]). This assignment is due on Tuesday 24 Feb. in class.

4 THE FOURTH ASSIGNMENT. 1. Let I 1,...,I n,... be disjoint subintervals (closed, open, or half-open). Assume I n = I, where I is also an interval. Denote by m(j) the length of an interval J. Prove: m(i) = m(i n ). 2. Let A be an arbitrary set, and let A 1,..., A n,... be subsets of A. Denote B n = A \ A n. Prove: A \ ( A n ) = B n ; A \ ( A n ) = B n. 3. For two sets A, B, write A B = (A \ B) (B \ A). For any three sets, A, B, C, prove: A B (A C) (B C). 4. Consider a sequence of sets E 1,...,E n,.... For any positive integer k, denote A k = E n ; B k = E n. Consider now the sets n=k k=1 Is it true that A B? that B A? n=k A = A k ; B = B k. 5. Give an example of a set X, a collection of its subsets A, such that if A, B A then also A B A, and an additive measure on A which does not extend to the ring generated by A. [Remark: Naturally, the collection A cannot be a semiring, and the point of the problem is to show that the second axiom of a semiring is essential). 6. Prove (in the context of abstract measures on semirings) that a countable union of sets of outer measure zero has outer measure zero. k=1 7. Let I n be subintervals of the unit interval such that m(i n ) <.

5 Let A be the set of points belonging to infinitely many of these subintervals: A = {x [0, 1] n N N > n : x I N }. Prove that A is a set of measure zero. For the problems below, recall that an open set in a metric space is one that with its every point contains a ball of positive radius around that point; that a closed set is one that contains all its accumulation points (where a point x is an accumulation point of a set A if every ball centred at x intersects A). 8. Prove that the union of any family of open sets is open; that the intersection of any family of closed sets is closed. 9. Prove that the complement to an open set is closed; the complement to a closed set is open. 10. Is it true that a finite subset of a metric space is always closed? 11. Prove that the intersection of a finite family of open sets is still open. Show that the intersection of an infinite family of open sets need not be open. 12. Prove that the union of a finite family of closed sets is still closed. Show that the union of an infinite family of closed sets need not be closed. 13. Prove that a finite union of sequentially compact sets is sequentially compact. 14. Prove that the intersection of any family of sequentially compact sets is sequentially compact. This assignment is due on Tuesday 17 Feb. in class. Previous assignments on next page.

6 THE FIRST ASSIGNMENT. Feller, Problems on handout: p.53, no.1 8; p.54, no. 9, 10 [10 is not graded]; p.58, no.5; p.59, no.15 and no.17. This assignment is due on Thursday 22 Jan. in class. THE SECOND ASSIGNMENT. Feller, Problems on handout: p.54, no. 11, 12, 15, 17, 19; p.56, no. 39, 40 [in 40, coins are indistinguishable, as are dice]; p.61, no. 1; p.62, no.8, 10, 11 [in 11, use (12.11), not (12.10)]. This assignment is due on Thursday 29 Jan. in class. THE THIRD ASSIGNMENT. Feller, Problems on handout: p.55, no.27, 28; p.61, no. 2-6; p.63, no. 18. Gnedenko, p.59, no.1; p.60, no. 2,3. Also, derive the formula of no. 16, p. 61 in Gnedenko. This assignment is due on Thursday 5 Feb. in class.