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 inherent reason to require any more than a measure on E for the classical probability and stochastic framework to work out.

It's not clear to me how the development in this paper provides significant benefits beyond what the classical definitions in probability and stochastic already produce. Could you clarify?