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Math Help - Countably additive for special set

  1. #1
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    Countably additive for special set

    Dears;
    Could you help me to prove the measure space(in the attachment) hold the countably additive?

    Countably additive for special set-s.png
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  2. #2
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    Re: Countably additive for special set

    Best Regards
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  3. #3
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    Re: Countably additive for special set

    and the countably additive
    Countably additive for special set-s2.png
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  4. #4
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    Re: Countably additive for special set

    There exist two disjoint sets with countable complements...
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    Re: Countably additive for special set

    Quote Originally Posted by emakarov View Post
    There exist two disjoint sets with countable complements...
    ]
    Thank you

    That is right But how can I find the measure specially at the right hand side of the countably additive
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  6. #6
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    Re: Countably additive for special set

    You can take E_1 and  E_2 to be two disjoint sets with countable complements and  E_i=\emptyset for i > 2. Then the left-hand side of the definition of countably additive measure is 1 while the right-hand side is 2.
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  7. #7
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    Re: Countably additive for special set

    Quote Originally Posted by emakarov View Post
    You can take E_1 and  E_2 to be two disjoint sets with countable complements and  E_i=\emptyset for i > 2. Then the left-hand side of the definition of countably additive measure is 1 while the right-hand side is 2.
    but they should be equal
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  8. #8
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    Re: Countably additive for special set

    I think this function is not a measure because it is not additive.
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