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

SOLVED:. Let X be a random variable. (i) Give an example of a probability space (Ω, ℱ, ℙ), a random variable X defined on this probability space, and a function f so that the σ-algebra

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