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| A sigma |
A '''sigma-algebra''' X over a ] E is a family of subsets of E with the following properties: | ||
| 1) E is in X, | 1) E is in X, | ||
| 2) If S is in X then so is the complement of X. | 2) If S is in X then so is the complement of X. | ||
| 3) If S<sub>1</sub>, S<sub>2</sub>, S<sub>3</sub>, ... is a sequence in X then their countable union must also be in X. | 3) If S<sub>1</sub>, S<sub>2</sub>, S<sub>3</sub>, ... is a sequence in X then their countable union must also be in X. | ||
| ⚫ | Examples: If E is any set, then each of these two families is a sigma-algebra: {0,E} (the trivial sigma-algebra), {S;S is a subset of E} (the power set). | ||
| If {X<sub>a</sub>} is a family of sigma-algebras, then the intersection of all X<sub>a</sub> is also a sigma-algebra. | |||
| If F is an arbitrary family of sets in E then we can form a special sigma-algebra from F, called the sigma-algebra generated by F. We denote it by σ(F) and define it as follows. | |||
| First note that there is a sigma-algebra that contains F, namely the power set. | |||
| Let G be the family of sigma-algebras containing F (that is, a sigma-algebra X over E is in G if and only if F is a subset of X.) | |||
| Then we simply define σ(F) to be the intersection of all sigma-algebras in G. | |||
| ⚫ | This leads to the most important example: the ] sigma-algebra in any ] is the sigma-algebra generated by the open sets (or, equivalently, by the closed sets). | ||
| ⚫ | Examples: If E is any set, then each of these two families is a sigma |
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| It is important to note that this sigma-algebra is not, in general, the power set. | |||
| For a non-trivial example, see the ]. | |||
| If F is an arbitrary family of sets in E then we can form a special sigma algebra from F, called the sigma algebra generated by F. We denote it by σ(F) and define it as follows. First note that there is a sigma algebra that contains F, namely the power set. Let G be the family of sigma algebrae containing F (that is, a sigma algebra X over E is in G if and only if F is a subset of X.) Then we simply define σ(F) to be the intersection of all sigma algebrae in G. | |||
| ⚫ | This leads to the most important example: the ] sigma |
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Revision as of 20:58, 6 February 2002
A sigma-algebra X over a set E is a family of subsets of E with the following properties:
1) E is in X,
2) If S is in X then so is the complement of X.
3) If S1, S2, S3, ... is a sequence in X then their countable union must also be in X.
Examples: If E is any set, then each of these two families is a sigma-algebra: {0,E} (the trivial sigma-algebra), {S;S is a subset of E} (the power set). If {Xa} is a family of sigma-algebras, then the intersection of all Xa is also a sigma-algebra.
If F is an arbitrary family of sets in E then we can form a special sigma-algebra from F, called the sigma-algebra generated by F. We denote it by σ(F) and define it as follows. First note that there is a sigma-algebra that contains F, namely the power set. Let G be the family of sigma-algebras containing F (that is, a sigma-algebra X over E is in G if and only if F is a subset of X.) Then we simply define σ(F) to be the intersection of all sigma-algebras in G.
This leads to the most important example: the Borel sigma-algebra in any topological space is the sigma-algebra generated by the open sets (or, equivalently, by the closed sets). It is important to note that this sigma-algebra is not, in general, the power set. For a non-trivial example, see the Vitali set.