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I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for describing the same thing, or are there fundamental differences?

At the bottom of the Probability Space page, it says that a prodability measure is a probability density of a random variable. However, near the top, it says, "The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example." Which suggests two me that they are two competing approaches/theories.

Can anyone explain to me how these terms and ideas fit together -- What are the primary conceptual differences between the two and the advantages/disadvantages of each?

I know that this question probably won't make sense to someone who actually understands what the terms really mean -- so please try to imagine a beginner who is just trying to make sense of the field, and understand why there are different terms that seem to apply to the same concepts.

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Can you post the links you have been reading? –  André Caldas Oct 13 '11 at 20:30
    
@Andre: I just posted a couple of them. –  Angada Oct 13 '11 at 21:49
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1 Answer

up vote 7 down vote accepted

A probability space $P$ is a measurable space $(P, \mathcal{P})$ (where $P$ is a set and $\mathcal{P}$ is a $\sigma$-algebra of subsets of $P$) equipped with a probability measure $\mu : \mathcal{P} \to \mathbb{R}_{\ge 0}$.

Given a probability space $P$, its algebra of random variables is the algebra of measurable functions $P \to \mathbb{R}$, where $\mathbb{R}$ is equipped its usual Borel $\sigma$-algebra. So no, these two concepts are not identical.

Every random variable $X$ induces a probability distribution over $\mathbb{R}$ as follows: we say that $$\mathbb{P}(X \in A) = \mu(X^{-1}(A))$$

where $A$ is a Borel subset of $\mathbb{R}$. For example, $$\mathbb{P}(a \le X \le b) = \mu(X^{-1}([a, b])).$$

The expected value $\mathbb{E}(X)$ of a random variable $X$ is its Lebesgue integral over $P$ as a function $P \to \mathbb{R}$. The higher moments $\mathbb{E}(X^p)$ are the expected values of the random variables $X^p$.

While the standard approach to rigorous probability emphasizes the probability space, it is possible to do probability in a way which instead emphasizes the algebra of random variables; some hints of what this looks like can be found in the beginning of Terence Tao's notes on free probability.

To really understand how this works it would be enormously helpful for you to learn a tiny bit of algebraic geometry. No, really! This is generally when one is first introduced to the notion that "algebra is dual to geometry" and the idea of studying a probability space by studying its algebra of random variables is very much in this vein.


The use of the word "algebra" in the term "$\sigma$-algebra" should not be confused with the use of the word "algebra" in the term "algebra of random variables." The latter refers to an algebra over a field (in this case $\mathbb{R}$), while the former refers to a collection of subsets with certain properties. The "algebra" in "$\sigma$-algebra" should be understood by analogy with Boolean algebra, where it simply indicates that intersection and union have algebraic properties which resemble those of addition and multiplication (but this should not be taken too seriously, at least for now, since intersection and union both distribute over each other!).

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The link you posted is extremely helpful and perfectly answers the question I was trying to ask. Thank you! –  Angada Oct 13 '11 at 22:11
    
You mean the notes on free probability? Yes, they were quite a revelation for me as well. –  Qiaochu Yuan Oct 13 '11 at 22:15
    
Yes. The 2nd-4th paragraphs directly answer my question (and maybe that makes it more clear what I was trying to ask). The rest is going to take some time to process, but I look forward to mulling over it. –  Angada Oct 13 '11 at 22:19
    
And, the stuff you just added is really helpful, too. Thank you! Especially the sigma algebra part. I wish I could double upvote! –  Angada Oct 13 '11 at 22:20
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