a). i) Define a field (also known as algebra) as used in the measure and probability theory. State the conditions that a given filed say A must meet in order to be a field. (4 marks) ii) Consider a field denoted by A and let A, A,..., A, E A. Using the ideas in (i) above, show that U1 A; = 1 A; A. (5 marks) iii) Suppose A is a class of sets containing 2 and satisfies Show that A is a field. A, B E A implies A \ B = AB A. b). Consider a probability measure defined as (S,F,P). Required: i) Explain each of the elements defined in the above space (S,F,P). (5 marks) (3 marks) ii) Consider A 22, explain all the properties of P. Use mathematical expressions. (3 marks) iii) If two dice are rolled once and we are interested in the events where the two numbers of that show up are equal (A), their sum are odd (A2),, their sums are 13 (A3). Apply the concept of a probability measure to come up with 2, F and P respectively for this experiment. (6 marks) c). Suppose, Y, Y2,... is a sequence of random variables with E(Y) and var (Yn) 0. Show that Y in probability. (4 marks)