Exercise 2.9. Let X be a random variable. (i) Give an example of a probability space...

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Exercise 2.9. Let X be a random variable. (i) Give an example of a probability space (, F,P), a random variable X defined on this probability space, and a function f so that the -algebra generated by f(x) is not the trivial -algebra {0, 1} but is strictly smaller than the -algebra generated by X. (ii) Can the -algebra generated by f(x) ever be strictly larger than the -algebra generated by X? Exercise 2.9. Let X be a random variable. (i) Give an example of a probability space (, F,P), a random variable X defined on this probability space, and a function f so that the -algebra generated by f(x) is not the trivial -algebra {0, 1} but is strictly smaller than the -algebra generated by X. (ii) Can the -algebra generated by f(x) ever be strictly larger than the -algebra generated by X?