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Wikipedia gives for Random Variable -- or more specifically, Random Element -- what seems to be a reasonably standard definition:

X: Ω -> E

where Ω is a space of outcomes and E is a set of values. Both Ω and E need to be measurable with respect to the probability measure on Ω and a measure on E.

While ℝ is both metric and ordered, and the metric generates a (Borel) measure, there is no in...

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random elements measure-theoretic probability
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