Consider the measurable space ((, A) and let X be defined by ni=1 i IAi or X

Consider the measurable space ((, A) and let X be defined by ∑ni=1 αi IAi or X = ∑ni=1 αi 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).