Thread: Countably additive for special set

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

Best Regards

and the countably additive

There exist two disjoint sets with countable complements...

Originally Posted by emakarov

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

You can take and to be two disjoint sets with countable complements and for i > 2. Then the left-hand side of the definition of countably additive measure is 1 while the right-hand side is 2.

Originally Posted by emakarov

You can take and to be two disjoint sets with countable complements and 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

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