[yshklarov]

I agree that good definitions are important, but I don't know if this is a fair criticism (even if the sentence were wrong) -- the purpose of this page is to clarify a common point of confusion, rather than to lay out a carefully formalized framework. Besides, the practice of introducing very restrictive definitions in introductory material is universal and, I would argue, pedagogically sound.

That said, while I appreciate and admire free textbooks published online, I think the exposition would be much improved if the author had a better sense of who he was writing for (the most common writing advice...).

And I take issue with the view that the sample space is where the random phenomenon lives, as it were. In my experience, it's more common to use the random variable itself to model the (observable aspect of) the random phenomenon, and for the sample space to be either a hidden (i.e., more abstract) aspect of the phenomenon or else a purely abstract formalism introduced only for ease of mathematical computation.

It would be helpful also to see some more context, especially historical (who introduced the concept of a sample space, and for what purpose?).