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
