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<math>\int_{\{|f_n|>M\}}|f_n|\leq\int_{(0,1)}|f_n-f|+\int_{\{|f_n|>M\}}|f|</math> | <math>\int_{\{|f_n|>M\}}|f_n|\leq\int_{(0,1)}|f_n-f|+\int_{\{|f_n|>M\}}|f|</math> | ||
| − | <math>Since \int_{(0,1)}|f_n-f|\to0(n\to\infty), it suffices to show\sup\int_{\{|f_n|>M\}}|f|\to0(M\to\infty)</math> | + | <math>Since \int_{(0,1)}|f_n-f|\to0(n\to\infty)</math>, it suffices to show that <math>\sup\int_{\{|f_n|>M\}}|f|\to0(M\to\infty)</math> |
Revision as of 08:53, 2 July 2008
- 1
$ \int_{\{|f_n|>M\}}|f_n|\leq\int_{(0,1)}|f_n-f|+\int_{\{|f_n|>M\}}|f| $
$ Since \int_{(0,1)}|f_n-f|\to0(n\to\infty) $, it suffices to show that $ \sup\int_{\{|f_n|>M\}}|f|\to0(M\to\infty) $
