November 21st, 2013, 04:55 PM  #1 
Newbie Joined: Sep 2013 Posts: 8 Thanks: 0  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? 
November 21st, 2013, 10:51 PM  #2  
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: measure Quote:
 
November 22nd, 2013, 08:55 AM  #3 
Newbie Joined: Sep 2013 Posts: 8 Thanks: 0  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..

November 22nd, 2013, 12:37 PM  #4 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  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 nonnegative real numbers (by additivity and nonnegativity of all measures) then exists and is finite. This is what permits us to pass the limit.

November 23rd, 2013, 11:02 AM  #5 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  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 nonnegativeness of all measures.


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