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In my textbook, one of the examples shows an $ \Omega $ that has two sigma algebras($ \mathcal F$ and $\mathcal B$). My only question is, how is it possible for there to be more than one sigma algebra? For example, suppose $ \Omega $ is {1,2}. Then the sigma algebra would be {$\Omega$, 1 , 2 , {1,2} , $\emptyset$}. What other ones could exist? Sorry if this seems like a simple question, I'm just having trouble grasping my head around it.

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    $\begingroup$ $\{\emptyset, \Omega\}$ is a $\sigma$-algebra as well. Why would you think that a set must have only one $\sigma$-algebra? $\endgroup$ – user384138 Jan 27 '17 at 15:56
  • $\begingroup$ I think I'm sort of understanding it now. Would the set {1,2} still be a $ \sigma $-algebra as well? $\endgroup$ – idude Jan 27 '17 at 15:58
  • $\begingroup$ No because a $\sigma$-algebra must have $\emptyset$ and the whole space as elements. $\endgroup$ – user384138 Jan 27 '17 at 15:59
  • $\begingroup$ Oh ok, I think I get it now. So in the same manner, {$\Omega$, $\emptyset$,{1,2}) would be a sigma algebra too? $\endgroup$ – idude Jan 27 '17 at 16:02
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    $\begingroup$ $\{1,2\}$ is just $\Omega$ (smh) $\endgroup$ – user384138 Jan 27 '17 at 16:02

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