# The axioms of sigma-agebras

1. Oct 14, 2006

### quasar987

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.

Last edited: Oct 14, 2006
2. Oct 14, 2006

### Hurkyl

Staff Emeritus
(2) applies to any sequence of elements of sigma. That includes sequences that don't contain a pair of complementary sets.

3. Oct 14, 2006

oh, right!