Talk:Probability axioms

From Wikipedia, the free encyclopedia

Would it be possibvle to rejig this in terms of Borel sets? Just a more elegant way of expressing it IMHO; here it's as if the defintion of sigma algebra comes out of probability theory. It might be moe confusing for the general reader however.


It's more confusing for me, anyway. The universe S is not necessarily a topological space, so what is a Borel set in this context?

 Fool 23:42 Mar 11, 2003 (UTC)

"Fool" is right. Borel sets are by definition members of the sigma-algebra generated by a topology. But there need not be any topology on a probability space. Or, at least, no topology is explicitly contemplated by the conventional Kolmogorovian definition.

I have moved this article to "Probability axioms" (plural!). Usually it is better to use the singular than the plural in the title of an article; "zebra" is better than "zebras". But in the case of this article, it is colossally silly. This is not about axioms as individual things; it is about systems of axioms, or, at least, about one particular system of axioms --- the one formulated by Kolmogorov. Michael Hardy 00:41 Mar 12, 2003 (UTC)

I think Cox's axioms should be stated here in addition to Kolmogorov's; if I'm not mistaken, the Kolmogorov axioms are derivable as theorems from Cox's -- which, again IINM, was Cox's point, that the accepted laws of probability are derivable from more basic assumptions. The one bit that doesn't carry over, if we start from Cox's axioms, is countable additivity -- IINM finite additivity derives from Cox's axioms but not countable additivity. Wile E. Heresiarch 15:30, 27 Dec 2003 (UTC)

I believe Axiom 1 should be stated simply as P(E) >= 0 rather than 0 <= P(E) <=1, since P(E) <=1 is actually a consequence of P(S) = 1, P(E) >= 0 and countable additivity.

[edit] Sigma-algebra

I'd like to rework this article to at least begin in a less technical manner, as I distinctly remember being confused by this when I learned it. But I'm not sure about the history here. I can understand why a modern mathematician working in ZFC needs this sort of thing to excommunicate non-measurable sets, but did Kolmogorov really care (or even know) about such things at all? Did his first axiom not apply to all events? Was his third axiom really countably additive and not just pairwise (therefore finitely) additive? -Dan (Fool) 03:47, 5 December 2005 (UTC)

Yes, his celebrated 1933 book does explicitly state the axiom as countable additivity. And he said that that, rather than just finite additivity, was merely for the sake of convenience. That Kolmogorov did NOT know about such things as sigma-algebras and non-measurable sets strikes me as implausible, but I haven't looked that closely. But after all, Kolmogorov's book appeared in 1933, so one should expect it to be quite modern in approach. Michael Hardy 22:11, 5 December 2005 (UTC)
Thanks. Actually, I figured it out, I was confusing him with Kronecker for some reason. Oops. -Dan 00:44, 6 December 2005 (UTC)

[edit] Proposed merge with probability theory

I propose this page has a merge with probability theory. Please add your comments to the proposal there. Thanks Andeggs 16:13, 24 December 2006 (UTC)

If someone can check that all of the relevant information has been moved to probability theory, we might be able to delete this page and replace it with a redirect? MisterSheik 17:12, 28 February 2007 (UTC)

I'm not for the merge anymore given that this page is linked to from probability space. And I'm not for folding probability space into probability theory given that the treatment at probability theory has a nice parallel structure with the other kinds of prob. theory, while the one at prob. space is more in depth, and clearer given no prior knowledge. Yeah, there's duplication, but there's also a lot of duplication of probability distribution, for example (which could really use a clean-up.) MisterSheik 17:41, 29 March 2007 (UTC)

[edit] Oversimplified?

From the discussion here it looks like this once discussed sigma-algebras and now doesn't. As it stands, this is just wrong - not every subset of omega can be assigned a probability. — ciphergoth 10:29, 11 November 2007 (UTC)