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This really is a conceptual question. As I have come across articles saying that sigma-algebra is generated by collection of sets and articles that say sigma-algebra is generated by a collection of random variables .

I know they mean the same thing but I would like to know how would connect the 2 ideas together . Anyone knows?

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2 Answers 2

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The σ-algebra generated by a collection of sets is the smallest σ-algebra containing all the given sets.

The σ-algebra generated by a collection $\mathcal F$ of functions is the smallest σ-algebra so that all those functions are measurable with respect to that σ-algebra. Equivalently, it is the σ-algebra generated by the collections of sets on the form $f^{-1}(-\infty,x]$ where $f\in\mathcal F$ and $x\in\mathbb{R}$. And that is the connection you are looking for.

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I assume you are aware that a random variable is just a measurable function on a probability space. I chose to use the slightly more general terminology in my answer; hope it did not confuse. –  Harald Hanche-Olsen Dec 1 '12 at 18:50
    
how would you call the form f^(-1) (-infinity, x] you mentioned up there??? Is there a particular name to it? –  user1769197 Dec 1 '12 at 19:09
1  
It is called the inverse image. More generally, $f^{-1}(A)=\{x\colon f(x)\in A\}$. –  Harald Hanche-Olsen Dec 1 '12 at 19:29

Yes, it is the same: A sigma algebra generated by a collection of random variables (say, real-valued) is the smallest sigma algebra that makes these random variables measurable: that means that it is the sigma algebra generated by the pre-images of any interval of the real line (with rational boundary points, if you prefer a countable generating collection).

Vice versa, on can say that a sigma algebra generated by a family of sets is the smallest sigma algebra which makes all the characteristic functions (i.e. the functions which identify these sets by being 1 inside such a set, and 0 outside) of these sets measurable.

You can find such elementary measuretheoretic concepts in any introductory textbook about measure theory. If your intention is to go deeper into math, I personally recommend the intro to measure theory by Paul Halmos, but I am sure there is plenty other great literature.

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