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October 14th, 2013, 02:10 AM   #1
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show that [latex]\mu[/latex] is a measure

let be a sigma algebra where
how to show that is a measure, where is a measure on

?
countable unions?
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October 14th, 2013, 03:21 AM   #2
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Re: show that [latex]\mu[/latex] is a measure

mu is a measure on M, so mu(empty) = 0

I give a proof for a union of two disjoint sets. I hope you can extend the proof for countable pairwisely disjoint sets.
I will detone the new measure by nu.
We have to show that nu(EuF u E'uF') = nu(EuF) + nu(E'uF'), where EuF and E'uF' are disjoint sets. The left side is equal with mu(E) + mu(E') by definiton of nu. The right hand side
nu(EuF u E'uF') = nu(EuE' u FuF') = nu(EuE' u F'') = mu(EuE') = mu(E) + mu(E')
where F'' = FuF' < NuN' = N''.
So the two sides are equal.
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October 14th, 2013, 03:30 AM   #3
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Re: show that [latex]\mu[/latex] is a measure

ah I see it now. Thank you!
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