> In the standard foundations of probability theory, as laid out by Kolmogorov, we can then model these events and random variables by introducing a sample space (which will be a probability space) to capture all the ambient sources of randomness; events are then modeled as measurable subsets of this sample space, and random variables are modeled as measurable functions on this sample space.
This matches the definitions of Wikipedia pretty closely: "A probability space consists of three parts: A sample space [...], A set of events [...], The assignment of probabilities to the events".
So either you misunderstood that sentence or both Wikipedia and Terrence Tao are wrong.
The sample space is not the probability space. The probability space has a sample space though, which is what the Wikipedia definition says, but not what Tao literally said. I think it is pretty clear what Tao meant though.
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)".
That's not good: We don't
want to have to use words
such as endowed and structure
that are not precisely
defined in the context and, thus,
are conceptually fuzzy.
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".
This matches the definitions of Wikipedia pretty closely: "A probability space consists of three parts: A sample space [...], A set of events [...], The assignment of probabilities to the events".
So either you misunderstood that sentence or both Wikipedia and Terrence Tao are wrong.