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Thread: Sigma-algebra

  1. #1
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    Sigma-algebra

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

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

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

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

    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 $\displaystyle \mathbb{Q}=(-\infty,+\infty)\cap \mathbb{Q}$ so, $\displaystyle \mathbb{Q}\in A$ .
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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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