Let $R$ be the smallest $\sigma$-algebra containing all compact sets in $\mathbb R^n$. I know that based on definition the minimal $\sigma$-algebra containing the closed (or open) sets is the Borel $\sigma$-algebra. But how can I prove that $R$ is actually the Borel $\sigma$-algebra?
is the smallest $\sigma$-algebra containing all compact sets the Borel $\sigma$-algebra
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real-analysis
measure-theory
compactness
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0In the second sentence, do you want to say _open_ sets? – 2012-09-05
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1I think the question should say "_the_ Borel $\sigma$-algebra" – 2012-09-05
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0@Dylan Moreland that is another definition of it – 2012-09-05
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1@Ana Well, that's what the exercise shows :) But what is written seems like a tautology. – 2012-09-05