Measure Theory/Probability.

Solve the following:

6.6 (a) Let (9,13), (@,A) and (AC) be measurable spaces, and let f: 9 > 9 and g : 9 > A be measurable functions. Show that g o f : 9 > A is measurable. (b) Let (9,13) and (@,A) be measurable spaces and f, g: Q ~> 9 measurable functions. Prove that (f, g): Q ~> G) x 8 is measurable With respect to the product a-algebra (the a-algebra generated by A X A). (c) Using (a) and (b)7 show that if X and Y are random variables then X iY, X - Y, X/Y (Y $ 0) are random variables