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- Jun 22, 2012

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I need help with the proof of Lemma 7.4.6 ...

Lemma 7.4.6 and its proof read as follows:

In the above proof by Lindstrom we read the following:

" ... ... Since this holds for any number \(\displaystyle a\) less than \(\displaystyle b\) and any number \(\displaystyle m\) less than \(\displaystyle \mu (B)\), we must have \(\displaystyle \lim_{ n \to \infty } \int_B f_n d \mu \geq b \mu (B)\) . ... ... "

I need help in order to show, formally and rigorously, that \(\displaystyle \lim_{ n \to \infty } \int_B f_n d \mu \geq b \mu (B)\) ... ...

My thoughts are that we could assume that \(\displaystyle \lim_{ n \to \infty } \int_B f_n d \mu \lt b \mu (B)\) ... ... and proceed to demonstrate a contradiction ... but I'm not sure how to formally proceed ... ...

Help will be much appreciated ...

Peter

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Readers of the above post may be assisted by access to Lindstrom's introduction to the integration of simple functions ... so I am providing access to the relevant text ... as follows:

Hope that helps ...

Peter