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| + | I believe 5(b) is false, as demonstrated by the counterexample <math>f(x)=\sum n\chi_{[n,n+\frac{1}{n^3}]}</math>. By Lebesgue's Differentiation Theorem, <math>f^*(x)\geq f(x)</math> a.e. So <math>f^*(x)</math> is unbounded outside any compact set. -pete | ||
Latest revision as of 11:01, 25 July 2008
Please post solutions here.
I believe 5(b) is false, as demonstrated by the counterexample $ f(x)=\sum n\chi_{[n,n+\frac{1}{n^3}]} $. By Lebesgue's Differentiation Theorem, $ f^*(x)\geq f(x) $ a.e. So $ f^*(x) $ is unbounded outside any compact set. -pete
