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Math Help - 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 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 = A^{c} then A =  \mathbb{R} = \Omega which belongs to F
    b) WLOG if A is finite let B = A^{c} is uncoutnable but 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 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 = A^{c} then A =  \mathbb{R} = \Omega which belongs to F
    b) WLOG if A is finite let B = A^{c} is uncoutnable but B^{c} is fintire thus B belongs in F.
    c) a union of finite sets is finite thus belong in F
    For each n\in\mathbb{Z}^+ is it true that \{n\}\in\mathcal{F}~?

    Is it true that \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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