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A sigma algebra over a set $X$ is defined to be closed under complement and countable union and to contain the unversal set.

Given a class of subsets $C$ of $X$, we can generate the smallest sigma algebra $\sigma{C}$ containing $C$. The most direct way is to add complements of current member in $C$ to $C$, add the countably unions of the members in $C$, and repeat and so on.

There are other ways to construct a sigma algebra from a class $C$ of subsets. For example, when $X = \mathbb{R}^n$, we first start with the semi-ring of all the half-open half-cloed rectangles, then generate its ring of subsets, and then generate a field of subset, and then generate the sigma algebra.

I can't see why the second way is less complicated than the first one.

I was wondering why the second construction is preferred over the first one? Is it because the first approach is to complicated?

Thanks and regards!

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The first way you described is not that precise. One rather considers all possible $\sigma$-algebras containing $C$ and defines $\sigma(C)$ as their intersection. As Michael has mentioned, generating $\sigma$-algebras is very useful for constructing measures. The described way is barely constructive, so one often uses extension theorems that show existence+uniqueness of the extension of pre-measures (or similar objects) defined on simpler structures (algebras, semi-rings) to $\sigma$-algebras. –  Ilya Jan 14 '13 at 9:23

2 Answers 2

The approach of semi-ring, to my knowledge is very strict and does not easily generalize to abstract measure spaces. I read Halmo's book 4-5 years ago and my impression is much of the theory is constructed very tightly like Artin's approach to Galois theory. While this had its advantages, for practical purpose I feel the $\sigma$-algebra approach is more flexible. None of the modern introductory measure theory texts I read/used used semi-ring very much (Folland, Tao, Rudin, Stein, Kolmogorov). I did not read Lang's book but I did not notice semi-ring from quick skimming. So I suspect the theory is now either outdate or reserved to some special situations where one need such a tight, strict theory.

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Usually, we want to eventually construct a measure on our $\sigma$-algebra. This is most often done using the Carathéodory's extension theorem. One starts with a countable additive set function on a semi-ring and extends it to a measure on the generated $\sigma$-algebra. It is nice to have to verify countable additivity only on a small family of sets, so it is sometimes practical to work with semi-rings. In particular, the family of measurable rectangles forms only a semi-ring. If $(X_i,\mathcal{X_i})$ is a family of measurable spaces, a measurable rectangle is an element of the form $\prod_i B_i$ such that $B_i\in\mathcal{X}_i$ for al $i$ and $B_i=X_i$ is for all but finitely many $i$. Measurable rectangles generate the product $\sigma$-algebra.

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