Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
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

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.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.