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Math Help - Borel signma fields

  1. #1
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    Borel signma fields



    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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  2. #2
    Super Member girdav's Avatar
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    Re: Borel sigma fields

    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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