Borel signma fields

**Helpmeplease1** · December 12th 2011, 11:52 AM

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

**girdav** · December 12th 2011, 12:13 PM

Denote $\mathcal C$ the collection of $(a,b)$ , $-\infty<a<b<+\infty$ , $\mathcal C_1$ the collection of $[a,b]$ , $-\infty<a<b<+\infty$ , etc... Since $[a,b]=\bigcap_{n\geq 1}\left(a-\frac 1n,b+\frac 1n\right)$ , each element of $\mathcal C_1$ is in the $\sigma$ -algebra generated by $\mathcal C$ , so $\mathcal B_1(\mathbb R)\subset\mathcal B(\mathbb R)$ . Use arguments of this type to get the other inclusions.

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