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November 21st, 2013, 05:55 PM   #1
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measure

Regarding this statement: http://i.imgur.com/F9YyQKi.png

I am having trouble incorporating in my proof. I understand why this must be true, but wouldn't the statement also still be true as long as for any k? Either way, I still don't know how to use it..

When trying to prove it Ive started by saying


Obviously ive made some mistake along the way because my proof doesn't use , can anyone point me in the right direction?
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November 21st, 2013, 11:51 PM   #2
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Re: measure

Quote:
Originally Posted by DuncanThaw
but wouldn't the statement also still be true as long as for any k?
- you are right, this is exactly what you need. If were infinite, then you could start from the for which because the intersection of A_m's from m = 1 to infinity and from m=k to infinity are the same. The reason you're told that is just so that you won't get a result like "".
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November 22nd, 2013, 09:55 AM   #3
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Re: measure

Thanks for your response, can you point out the error in my proof? I did not use the assumption that so it must be wrong in some way..
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November 22nd, 2013, 01:37 PM   #4
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Re: measure

What I suggest is that you justify the last step you took, where the limit has been passed into the measure without explanation. You should justify doing this because some measures aren't exactly continuous functions on the measure space. Because is a decreasing sequence of non-negative real numbers (by -additivity and non-negativity of all measures) then exists and is finite. This is what permits us to pass the limit.
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November 23rd, 2013, 12:02 PM   #5
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Re: measure

DuncanThaw, alas I suggest that you write instead of to completely avoid passing the limit into the measure (I'm not at all comfortable with doing this). Still justify finiteness of the measure due to -additivity and non-negativeness of all measures.
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