Читать книгу Probability and Statistical Inference - Robert Bartoszynski - Страница 42

1 1.4.1 Let be a countable partition of ; that is, for all , and . Let . Find .

2 1.4.2 Assume that John will live forever. He plays a certain game each day. Let be the event that he wins the game on the th day. (i) Let be the event that John will win every game starting on January 1, 2035. Label the following statements as true or false: (a) . (b) . (c) . (d) . (ii) Assume now that John starts playing on a Monday. Match the following events through with events through : John loses infinitely many games. When John loses on a Thursday, he wins on the following Sunday. John never wins on three consecutive days. John wins every Wednesday. John wins on infinitely many Wednesdays. John wins on a Wednesday. John never wins on a weekend. John wins infinitely many games and loses infinitely many games. If John wins on some day, he never loses on the next day.

3 1.4.3 Let be distinct subsets of . (i) Find the maximum number of sets (including and ) of the smallest field containing . (ii) Find the maximum number of sets in this field if . (iii) Answer (ii) if . (iv) Answer (ii) if . (v) Answer (i)–(iv) for a ‐field.

4 1.4.4 For let . Consider a sequence of numbers satisfying for all , and let . (i) Find and . (ii) Find conditions, expressed in terms of , under which exists, and find this limit. (iii) Define and . Answer questions (i) and (ii) for sequence .

5 1.4.5 Let be the set of all integers. For , let be the number of elements in the intersection . Let be the class of all sets for which the limitexists. Show that is not a field. [Hint: Let and { all odd integers between and and all even integers between and for . Show that both and are in but .]

6 1.4.6 Let . Show that the class of all finite unions of intervals of the form and , with possibly infinite or (intervals of the form etc.) forms a field.