[vcxy]

I'm an American mathematician and have always allowed the codomain of a random variable to be any measurable space. I haven't noticed anyone mention random elements. I don't work in probability though, so maybe people directly in the field care more.

[yshklarov]

From what I saw as a recent grad student in probability, most texts do define a random variable to necessarily map into the reals, or the extended reals or perhaps a subset thereof, or occasionally the complex numbers, and the more general concept is a "random element" (when a more specific term is called for, there are "random vectors", "random graphs", "random processes", etc.). But this is certainly not universal even within probability. In any case, I don't believe it matters much -- it's hard to see how a mix-up here might cause any real confusion, though as always it is annoying that there isn't a common convention.

[semi-extrinsic]

> it's hard to see how a mix-up here might cause any _real_ confusion

I really appreciate comments like this.