(Solution) - Consider the measurable space A and let X be -(2025 Original AI-Free Solution)

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Consider the measurable space ((, A) and let X be defined by ?ni=1 ?i IAi or X = ?ni=1 ?i IAi, where ai ( ( are distinct for all i and {A1,..., An}or {Ai, i ? 1} are partitions of (. Then show that X is a r.v. (a simple r.v. and an elementary r.v., respectively) if and only if the partitions are measurable (i.e., Ai ( A for all i).