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