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> 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 spa
by olalonde 11y ago
> 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.
- d9fb698e010974b 11y agoThe 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.
- graycat 11y ago" ... a sample space (which will be a probability space)" is, in a word, wrong. The sample space will be part of a probability space but will not "be a probability space".
- elliptic 11y agoNonsense. Tao is using perfectly idiomatic language here - "The sample space will be a probability space (once we have endowed it with some additional structure)".
- graycat 11y agoThat'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". The definitions I gave are the accepted ones.