probability
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 a-algebra generated by f(X) is not the trivial σ-algebra {θ, 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 a-algebra generated by X?
probability