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In his notes on free probability, Terence Tao describes a general approach to non-commutative probability which prioritizes the algebra of random variables above the sample space; I find this conceptually appealing. I would be interested in finding a reference which develops this theory, even if only in the commutative case, to the point where one can reproduce standard probabilistic and measure-theoretic results (e.g. the SLLN, the central limit theorem), and I would also be interested in applications to a measure-space-free statement and proof of an ergodic theorem.

Motivation: A problem on a recent problem set of mine has convinced me that the measure-theoretic and probabilistic apparatus I'm familiar with would be more flexible if I didn't have to think about sample spaces. I am also interested in having a probabilistic language that adapts to quantum probability more readily.

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What's wrong with the canonical sample space for a fixed set of random variables though? (I'm also curious which specific problems you're thinking of. The only question where the choice of sample space seemed relevant was the one about a stationary sequence of random variables.) –  Zhen Lin Jan 22 '11 at 22:17
    
@Zhen: yes, I was thinking of the one about a stationary sequence of random variables. I ended up using the canonical sample space, but my point is that the transformation I was trying to define was already naturally defined on the algebra itself, and I'd like to do measure theory in a framework where I don't have to relate this to a transformation of an auxiliary set which is ultimately of secondary importance. –  Qiaochu Yuan Jan 22 '11 at 23:10
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4 Answers

A good book:

Lectures on the combinatorics of free probability-A. Nica and R. Speicher

See the following article and the references therein for information on noncommutative ergodic theory:

Noncommutative maximal ergodic theorems-M Junge and Q. Xu

http://arxiv.org/PS_cache/math/pdf/0505/0505308v2.pdf

(see also other articles of these authors)

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An approach based on commutative $C*$ algebras is given in the following article by Patra and Braunstein. The article includes examples of the construction of some known random variables. Apart from the motivation that this approach is generalizable to the quantum case, the authers indicate that it provides some additional computational tools. However, the article basically treats the special case of bounded random variables. The construction in the unbouded case is harder. I don't know of a reference for the general theory, but the following article by: G. F. Vincent-Smith treats the case of Gaussian stochastic processes along the same lines.

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Some of the answers of Dmitri Pavlov, particularly this one will be helpful to you. Segal's 1965 Algebraic integration theory is also a good place to look; he walks you through for example how to view a measure space as a "spectrum" of an algebra of random variables (e.g. the unnumbered theorem on page 432).

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I don't know if this is exactly what you are searching for, but maybe it is worth to take a look:

Probability Theory: The Logic of Science by. E. T. Jaynes.

http://bayes.wustl.edu/etj/prob/book.pdf

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How is this relevant (even approximately)? –  R Hahn Jan 23 '11 at 7:07
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