# Countably additive for special set

• May 16th 2012, 02:12 PM
raed
Dears;
Could you help me to prove the measure space(in the attachment) hold the countably additive?

Attachment 23886
• May 16th 2012, 02:13 PM
raed
Re: Countably additive for special set
Best Regards
• May 16th 2012, 02:15 PM
raed
Re: Countably additive for special set
Attachment 23887
• May 16th 2012, 03:13 PM
emakarov
Re: Countably additive for special set
There exist two disjoint sets with countable complements...
• May 17th 2012, 03:50 AM
raed
Re: Countably additive for special set
Quote:

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
• May 17th 2012, 07:33 AM
emakarov
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.
• May 17th 2012, 08:19 AM
raed
Re: Countably additive for special set
Quote:

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
• May 17th 2012, 08:21 AM
emakarov
Re: Countably additive for special set
I think this function is not a measure because it is not additive.