http://i44.tinypic.com/2j4o64m.png

I really don't understand what to do, i know a borel sigma field is the set with the smallest number of fields but i dont understand how to show this :(

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- Dec 12th 2011, 11:52 AMHelpmeplease1Borel signma fields
http://i44.tinypic.com/2j4o64m.png

I really don't understand what to do, i know a borel sigma field is the set with the smallest number of fields but i dont understand how to show this :( - Dec 12th 2011, 12:13 PMgirdavRe: Borel sigma fields
Denote $\displaystyle \mathcal C$ the collection of $\displaystyle (a,b)$, $\displaystyle -\infty<a<b<+\infty$, $\displaystyle \mathcal C_1$ the collection of $\displaystyle [a,b]$, $\displaystyle -\infty<a<b<+\infty$, etc... Since $\displaystyle [a,b]=\bigcap_{n\geq 1}\left(a-\frac 1n,b+\frac 1n\right)$, each element of $\displaystyle \mathcal C_1$ is in the $\displaystyle \sigma$-algebra generated by $\displaystyle \mathcal C$, so $\displaystyle \mathcal B_1(\mathbb R)\subset\mathcal B(\mathbb R)$. Use arguments of this type to get the other inclusions.