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Generating the Borel-algebra from half-open intervals

  1. Dec 8, 2011 #1
    Hi everybody!

    I have been asked to show that the Borel-algebra can be generated from the set of half-open intervals of the form [a , b) where a<b.

    I know that the set of open intervals of the form (a,b) where a<b generates the Borel-algebra and thought I would go about showing that the to sets generates the same δ-algebra. But that has proven more difficult than I thought.

    Can anybody give me an direction?


    Any help is greatly appreciated!
    dane502
     
  2. jcsd
  3. Dec 8, 2011 #2

    micromass

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    Can you write the open intervals as a union of halfopen intervals?

     
  4. Dec 8, 2011 #3
    Thank you for your answer.

    No, the half-open intervals has to be on the form of [a,b), so any union of those half-open intervals (that is not disjoint) will also have form [a,b).
     
  5. Dec 8, 2011 #4

    micromass

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    Sure of that?
     
  6. Dec 8, 2011 #5

    micromass

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    Think of

    [tex]\bigcup [a-1/n,b)[/tex]
     
  7. Dec 8, 2011 #6
    Thank you. Somehow I missed that.
     
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