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Math Help - Sigma-algebra

  1. #1
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    Sigma-algebra

    Let A the collection of finite unions of sets of the form
    (a,b] \cap \mathbb{Q}, when -\infty \leqa<b \leq +\infty."

    a. Show that A is an algebra on \mathbb{Q}.

    b. Show that the \sigma-algebra generated by A is
    P( \mathbb{Q}) (the power set of \mathbb{Q}).

    c. Define \nu on A by \nu( \phi)=0 and \nu(C)= +\infty for C \neq phi. Show that \nu is a premeasure on A and that there is more than one measure on P( \mathbb{Q}) whose restriction to A is \nu.

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Sigma-algebra

    It would be convenient to show some work. What difficulties have you had?. For example \mathbb{Q}=(-\infty,+\infty)\cap \mathbb{Q} so, \mathbb{Q}\in A .
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  3. #3
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    Re: Sigma-algebra

    I have problems to prove (b); one of the implications is obvious because the sigma-algebra generated by A is contained in P(Q), but I don't know how to prove the other direction. Thanks.
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  4. #4
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    Re: Sigma-algebra

    I just solved the problem. It was quite simple :/. Sorry for your time.
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