| Revision as of 07:06, 8 March 2002 editZundark (talk | contribs)Extended confirmed users, File movers, Pending changes reviewers29,659 edits define "measurable space" here (hardly seems worth a separate article)← Previous edit | Revision as of 07:24, 8 March 2002 edit undoAxelBoldt (talk | contribs)Administrators44,502 edits Switching ''E'' and ''S'' since it is more common in Probability theory that wayNext edit → | ||
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| A '''σ-algebra''' (or '''σ-field''') ''X'' over a ] '' |
A '''σ-algebra''' (or '''σ-field''') ''X'' over a ] ''S'' is a family of ] of ''S'' which is closed under ] set operations; σ-algebras are mainly used in order to define ] on ''S''. The concept is important in ] and ]. | ||
| Formally, ''X'' is a σ-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 '' |
# If ''E'' is in X then so is the complement of ''E''. | ||
| # If '' |
# If ''E''<sub>1</sub>, ''E''<sub>2</sub>, ''E''<sub>3</sub>, ... is a sequence in ''X'' then their (countable) union is also in ''X''. | ||
| From 1 and 2 it follows that '' |
From 1 and 2 it follows that ''S'' is in ''X''; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections. | ||
| An ordered pair ('' |
An ordered pair (''S'', ''X''), where ''S'' is a set and ''X'' is a σ-algebra over ''S'', is called a '''measurable space'''. | ||
| === Examples === | === Examples === | ||
| If '' |
If ''S'' is any set, then the family consisting only of the empty set and ''S'' is a σ-algebra over ''E'', the so-called ''trivial σ-algebra''. Another σ-algebra over ''S'' is given by the full ] of ''S''. | ||
| If {''X''<sub>a</sub>} is a family of σ-algebras over '' |
If {''X''<sub>a</sub>} is a family of σ-algebras over ''S'', then the intersection of all ''X''<sub>a</sub> is also a σ-algebra over ''S''. | ||
| If ''U'' is an arbitrary family of subsets of '' |
If ''U'' is an arbitrary family of subsets of ''S'' 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 '' |
First note that there is a σ-algebra over ''S'' that contains ''U'', namely the power set of ''S''. | ||
| Let Φ be the family of all σ-algebras over '' |
Let Φ be the family of all σ-algebras over ''S'' that contain ''U'' (that is, a σ-algebra ''X'' over ''S'' 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 '' |
Then we define σ(''U'') to be the intersection of all σ-algebras in Φ. σ(''U'') is then the smallest σ-algebra over ''S'' 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 ] |
This leads to the most important example: the ] over any ] 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. | 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 σ-algebra is of importance: that of all ] sets. This σ-algebra contains more sets than the Borel |
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 algebra on '''R'''<sup>''n''</sup> and is preferred in ] theory. | ||
| === Measurable functions === | === Measurable functions === | ||
| If ''X'' is a σ-algebra over '' |
If ''X'' is a σ-algebra over ''S'' and ''Y'' is a σ-algebra over ''T'', then a ] ''f'' : ''S'' <tt>-></tt> ''T'' is called ''measurable'' if the preimage of every set in ''Y'' is in ''X''. A function ''f'' : ''S'' <tt>-></tt> '''R''' is called measurable if it is measurable with respect to the Borel σ-algebra on '''R'''. | ||
Revision as of 07:24, 8 March 2002
A σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. 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 E is in X then so is the complement of E.
- If E1, E2, E3, ... is a sequence in X then their (countable) union is also in X.
From 1 and 2 it follows that S is in X; from 2 and 3 it follows that the σ-algebra is also closed under countable intersections.
An ordered pair (S, X), where S is a set and X is a σ-algebra over S, is called a measurable space.
Examples
If S is any set, then the family consisting only of the empty set and S is a σ-algebra over E, the so-called trivial σ-algebra. Another σ-algebra over S is given by the full power set of S.
If {Xa} is a family of σ-algebras over S, then the intersection of all Xa is also a σ-algebra over S.
If U is an arbitrary family of subsets of S 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 S that contains U, namely the power set of S. Let Φ be the family of all σ-algebras over S that contain U (that is, a σ-algebra X over S 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 S 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 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 algebra on R and is preferred in integration theory.
Measurable functions
If X is a σ-algebra over S and Y is a σ-algebra over T, then a function f : S -> T is called measurable if the preimage of every set in Y is in X. A function f : S -> R is called measurable if it is measurable with respect to the Borel σ-algebra on R.