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.
How can we have multiple sigma algebras?
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probability
probability-theory
measure-theory
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1$\{\emptyset, \Omega\}$ is a $\sigma$-algebra as well. Why would you think that a set must have only one $\sigma$-algebra? – 2017-01-27
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0I think I'm sort of understanding it now. Would the set {1,2} still be a $ \sigma $-algebra as well? – 2017-01-27
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0No because a $\sigma$-algebra must have $\emptyset$ and the whole space as elements. – 2017-01-27
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0Oh ok, I think I get it now. So in the same manner, {$\Omega$, $\emptyset$,{1,2}) would be a sigma algebra too? – 2017-01-27
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1$\{1,2\}$ is just $\Omega$ (smh) – 2017-01-27
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0Yeah, sorry about that. It makes sense now, the textbook I had just explained it in a really odd manner. If you wish, you can post an answer, and I'll be happy to accept it – 2017-01-27
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0I'm glad I helped. Thanks, but I don't feel like writing an answer. – 2017-01-27