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Sigma-Algebra or Sigma-Field is a very important concept in Information theory. It has been used to represent all the possible information before a certain time. But what is sigma-algebra?
Given a Set Omega, a Sigma-Algebra is a class of subsets of Omega, which satisfies:
1. empty set and Omega are included;
2. if set A is included, then so is the complement of A;
3. if sets A and B are included, then so are the interseption and the union of them;
4. if a sequence A1...An... are included, then so are the limits of interseption and the union of them.
Following the four rules above, you can easily find out what is sigma-algebra and what is not.
Note that it is not necessary that all the subsets in Omega be included, but the completeness of intersection and union to infinity are essential.
The Borel Sigma Algebra is a special Sigma-Algebra, which includes all the open subsets of Omega!
It can be proved that, in fact, all the subsets of Omega are included in Borel Sigma Algebra.
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