- #1
Rasalhague
- 1,387
- 2
What is "complementation" wrt a sigma-algebra?
What does complementation mean here? Is the statement saying that the complement of a subset of X in [itex]\Sigma[/itex] must also be in [itex]\Sigma[/itex] for [itex]\Sigma[/itex] to qualify as the underlying set of a [itex]\sigma[/itex]-algebra?
[tex]A \in \Sigma \Rightarrow A \subset X[/tex]
and
[tex]A \in \Sigma \Rightarrow \enspace \{ x : x \in X, x \notin A \} \in \Sigma[/tex]
And is "collection" just a convenient synonym for set?
By definition, a [itex]\sigma[/itex]-algebra over a set X is a nonempty collection [itex]\Sigma[/itex] of subsets of X (including X itself) that is closed under complementation and countable unions of its members.
http://en.wikipedia.org/wiki/Σ-algebra
What does complementation mean here? Is the statement saying that the complement of a subset of X in [itex]\Sigma[/itex] must also be in [itex]\Sigma[/itex] for [itex]\Sigma[/itex] to qualify as the underlying set of a [itex]\sigma[/itex]-algebra?
[tex]A \in \Sigma \Rightarrow A \subset X[/tex]
and
[tex]A \in \Sigma \Rightarrow \enspace \{ x : x \in X, x \notin A \} \in \Sigma[/tex]
And is "collection" just a convenient synonym for set?