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Thread: is the set F = {A|A or A^c is finite} over R a sigma algebra?

  1. #1
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    is the set F = {A|A or A^c is finite} over R a sigma algebra?

    define $\displaystyle F = \{ A | A \mbox{ or } A^{c} \mbox{is finite} \} (\Omega = \mathbb{R}). \mbox{ Is F a } \sigma \mbox{-algebra over } \mathbb{R} ?$

    My anwser: Yes as
    a) if empty set = $\displaystyle A^{c}$ then A = $\displaystyle \mathbb{R} = \Omega$ which belongs to F
    b) WLOG if A is finite let B = $\displaystyle A^{c}$ is uncoutnable but $\displaystyle B^{c}$ is fintire thus B belongs in F.
    c) a union of finite sets is finite thus belong in F
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  2. #2
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    Quote Originally Posted by rosh3000 View Post
    define $\displaystyle F = \{ A | A \mbox{ or } A^{c} \mbox{is finite} \} (\Omega = \mathbb{R}). \mbox{ Is F a } \sigma \mbox{-algebra over } \mathbb{R} ?$

    My anwser: Yes as
    a) if empty set = $\displaystyle A^{c}$ then A = $\displaystyle \mathbb{R} = \Omega$ which belongs to F
    b) WLOG if A is finite let B = $\displaystyle A^{c}$ is uncoutnable but $\displaystyle B^{c}$ is fintire thus B belongs in F.
    c) a union of finite sets is finite thus belong in F
    For each $\displaystyle n\in\mathbb{Z}^+$ is it true that $\displaystyle \{n\}\in\mathcal{F}~?$

    Is it true that $\displaystyle \bigcup\limits_n {\left\{ n \right\}}\in\mathcal{F}~? $
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  3. #3
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    Yes clearly finite,
    No union is infinte and complement is uncountable infinte thus not in f

    Thank you
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