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Wiki says that a sigma algebra (or sigma field) is a subset [itex]\Sigma[/itex] of the powerset of some set X satisfying the following axioms
1) [tex]E\in \Sigma \Rightarrow E^c \in \Sigma[/tex]
2) [tex]E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma[/tex]
(where the index set I is countable)
Am I missing something or is axiom 2 equivalent to the much less complicated "2') [itex]X\in \Sigma[/itex]"? Cause for any element of [itex]\Sigma[/itex], since its complement is in [itex]\Sigma[/itex] also, the union of both is X itself. So 2) is satified as soon as 2') is. Conversely, 2) implies that X is in [itex]\Sigma[/itex] simply by taking an element of [itex]\Sigma[/itex] and its complement in the union.
1) [tex]E\in \Sigma \Rightarrow E^c \in \Sigma[/tex]
2) [tex]E_i \in \Sigma \ \ \forall i \in I \Rightarrow \bigcup_{i\in I}E_i \in \Sigma[/tex]
(where the index set I is countable)
Am I missing something or is axiom 2 equivalent to the much less complicated "2') [itex]X\in \Sigma[/itex]"? Cause for any element of [itex]\Sigma[/itex], since its complement is in [itex]\Sigma[/itex] also, the union of both is X itself. So 2) is satified as soon as 2') is. Conversely, 2) implies that X is in [itex]\Sigma[/itex] simply by taking an element of [itex]\Sigma[/itex] and its complement in the union.
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