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| − | Recall if <math>f\in L^1_{loc}, </math> the result for # | + | Recall if <math>f\in L^1_{loc}, </math> the result for #3a follows from Lebesgue differentiation theorem. |
Next if <math>f\notin L^1_{loc}</math> consider the following proof: | Next if <math>f\notin L^1_{loc}</math> consider the following proof: | ||
Revision as of 15:08, 9 July 2008
Recall if $ f\in L^1_{loc}, $ the result for #3a follows from Lebesgue differentiation theorem.
Next if $ f\notin L^1_{loc} $ consider the following proof: WLOG $ f\geq 0 $ by replacing $ f $ with $ |f|. $
Since $ f \notin L^1_{loc}, \exists x\in \mathbb{R}^n and \exists K\subset \mathbb{R}^n, K $ compact,s.t. $ \int_Kf=\infty $. Choose any $ y\in\mathbb{R}^n $. Then choose a cube $ Q\supseteq K $ centered at $ y $ which is possible since $ K $ compact implies $ K $ bounded. Then $ f^*(y)\geq \dfrac{\int_Qf}{|Q|} \geq \dfrac{\int_Kf}{|Q|}=\infty $, so we have the result, namely $ \{ f* \geq f \} =\mathbb{R}^n $
QED
