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| A '''sigma-algebra''' ''X'' over a ] ''E'' is a family of ] of ''E'' which is closed under ] set operations; sigma-algebras are mainly used in order to define ] on ''E''. The concept is important in ] and ]. | A '''σ-algebra''' ''X'' over a ] ''E'' is a family of ] of ''E'' which is closed under ] set operations; σ-algebras are mainly used in order to define ] on ''E''. The concept is important in ] and ]. | ||
| Formally, ''X'' is a sigma-algebra if and only if it has the following properties: | Formally, ''X'' is a σ-algebra if and only if it has the following properties: | ||
| # The empty set is in ''X'', | # The empty set is in ''X'', | ||
| # If ''S'' is in X then so is the complement of ''S''. | # If ''S'' is in X then so is the complement of ''S''. | ||
| # If ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ... is a sequence in ''X'' then their countable union is also in ''X''. | # If ''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ... is a sequence in ''X'' then their (countable) union is also in ''X''. | ||
| From 1 and 2 it follows that ''E'' is in ''X''; from 2 and 3 it follows that the sigma-algebra is also closed under countable intersections. | From 1 and 2 it follows that ''E'' is in ''X''; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections. | ||
| === Examples === | === Examples === | ||
| If ''E'' is any set, then the family consisting only of the empty set and ''E'' is a sigma-algebra over ''E'', the so- |
If ''E'' is any set, then the family consisting only of the empty set and ''E'' is a σ-algebra over ''E'', the so-called ''trivial σ-algebra''. Another σ-algebra over ''E'' is given by the ] of ''E''. | ||
| If {''X''<sub>a</sub>} is a family of sigma-algebras over ''E'', then the intersection of all ''X''<sub>a</sub> is also a sigma-algebra over ''E''. | If {''X''<sub>a</sub>} is a family of σ-algebras over ''E'', then the intersection of all ''X''<sub>a</sub> is also a σ-algebra over ''E''. | ||
| If ''U'' is an arbitrary family of subsets of ''E'' then we can form a special sigma-algebra from ''U'', called the ''sigma-algebra generated by U''. We denote it by σ(''U'') and define it as follows. | If ''U'' is an arbitrary family of subsets of ''E'' then we can form a special σ-algebra from ''U'', called the ''σ-algebra generated by U''. We denote it by σ(''U'') and define it as follows. | ||
| First note that there is a sigma-algebra over ''E'' that contains ''U'', namely the power set of ''E''. | First note that there is a σ-algebra over ''E'' that contains ''U'', namely the power set of ''E''. | ||
| Let Φ be the family of all sigma-algebras over ''E'' that contain ''U'' (that is, a sigma-algebra ''X'' over ''E'' is in Φ if and only if ''U'' is a subset of ''X''.) | Let Φ be the family of all σ-algebras over ''E'' that contain ''U'' (that is, a σ-algebra ''X'' over ''E'' is in Φ if and only if ''U'' is a subset of ''X''.) | ||
| Then we define σ(''U'') to be the intersection of all sigma-algebras in Φ. σ(''U'') is then the smallest sigma-algebra over ''E'' that contains ''U''; its elements are all sets that can be gotten from sets in ''U'' by applying a countable sequence of the set operations union, intersection and complement. | Then we define σ(''U'') to be the intersection of all σ-algebras in Φ. σ(''U'') is then the smallest σ-algebra over ''E'' that contains ''U''; its elements are all sets that can be gotten from sets in ''U'' by applying a countable sequence of the set operations union, intersection and complement. | ||
| This leads to the most important example: the ] sigma-algebra over any ] is the sigma-algebra generated by the open sets (or, equivalently, by the closed sets). | This leads to the most important example: the ] σ-algebra over any ] is the σ-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 whole power set. | It is important to note that this σ-algebra is not, in general, the whole power set. | ||
| For a non-trivial example, see the ]. | For a non-trivial example, see the ]. | ||
| On the ] '''R'''<sup>''n''</sup>, another sigma-algebra is of importance: that of all ] sets. This sigma |
On the ] '''R'''<sup>''n''</sup>, another σ-algebra is of importance: that of all ] sets. This σ-algebra contains more sets than the Borel σ-algebra. | ||
| === Measurable functions === | === Measurable functions === | ||
| If ''X'' is a sigma-algebra over ''E'' and ''Y'' is a sigma |
If ''X'' is a σ-algebra over ''E'' and ''Y'' is a σ-algebra over ''F'', then a ] ''f'' : ''E'' <tt>-></tt> ''F'' is called ''measurable'' if the preimage of every set in ''Y'' is in ''X''. A function ''f'' : ''E'' <tt>-></tt> '''R''' is called measurable if it is measurable with respect to the Borel σ-algebra on '''R'''. | ||
Revision as of 05:33, 6 March 2002
A σ-algebra X over a set E is a family of subsets of E which is closed under countable set operations; σ-algebras are mainly used in order to define measures on E. The concept is important in analysis and probability and statistics.
Formally, X is a σ-algebra if and only if it has the following properties:
- The empty set is in X,
- If S is in X then so is the complement of S.
- If S1, S2, S3, ... is a sequence in X then their (countable) union is also in X.
From 1 and 2 it follows that E is in X; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections.
Examples
If E is any set, then the family consisting only of the empty set and E is a σ-algebra over E, the so-called trivial σ-algebra. Another σ-algebra over E is given by the power set of E.
If {Xa} is a family of σ-algebras over E, then the intersection of all Xa is also a σ-algebra over E.
If U is an arbitrary family of subsets of E then we can form a special σ-algebra from U, called the σ-algebra generated by U. We denote it by σ(U) and define it as follows. First note that there is a σ-algebra over E that contains U, namely the power set of E. Let Φ be the family of all σ-algebras over E that contain U (that is, a σ-algebra X over E is in Φ if and only if U is a subset of X.) Then we define σ(U) to be the intersection of all σ-algebras in Φ. σ(U) is then the smallest σ-algebra over E that contains U; its elements are all sets that can be gotten from sets in U by applying a countable sequence of the set operations union, intersection and complement.
This leads to the most important example: the Borel σ-algebra over any topological space is the σ-algebra generated by the open sets (or, equivalently, by the closed sets). It is important to note that this σ-algebra is not, in general, the whole power set. For a non-trivial example, see the Vitali set.
On the Euclidean space R, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra.
Measurable functions
If X is a σ-algebra over E and Y is a σ-algebra over F, then a function f : E -> F is called measurable if the preimage of every set in Y is in X. A function f : E -> R is called measurable if it is measurable with respect to the Borel σ-algebra on R.