Nonsense. Tao is using perfectly idiomatic language here - "The sample space will be a probability space (once we have endowed it with some additional structure)".
Some intuitive overviews, clearly labeled as such, are fine and can be helpful, but "idiomatic language" just is not. Won't find such in the writings of W. Rudin, P. Halmos, J. Neveu, or any of a long list of authors of some of the best math books. In a good math or computer science journal, a reviewer or the editor would likely reject "idiomatic language".
The definitions I gave are the accepted ones.