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- I need help in order to fully understand Lindstrom's proof of Lemma 7.4.6 concerning the Lebesgue integration of an increasing sequence of simple functions ...
I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ...
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 ##a## less than ##b## and any number ##m## less than ##\mu (B)##, we must have ##\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 ##\lim_{ n \to \infty } \int_B f_n d \mu \geq b \mu (B)## ... ...My thoughts are that we could assume that ##\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
=================================================================================================================
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
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 ##a## less than ##b## and any number ##m## less than ##\mu (B)##, we must have ##\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 ##\lim_{ n \to \infty } \int_B f_n d \mu \geq b \mu (B)## ... ...My thoughts are that we could assume that ##\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
=================================================================================================================
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