1. ## Countably additive for special set

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

Best Regards

4. ## Re: Countably additive for special set

There exist two disjoint sets with countable complements...

5. ## Re: Countably additive for special set

Originally Posted by emakarov
There exist two disjoint sets with countable complements...
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Thank you

That is right But how can I find the measure specially at the right hand side of the countably additive

6. ## 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.

7. ## Re: Countably additive for special set

Originally Posted by emakarov
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

8. ## Re: Countably additive for special set

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