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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: | ||
WLOG <math>f\geq 0 </math> by replacing <math> f </math> with <math> |f|.</math> | WLOG <math>f\geq 0 </math> by replacing <math> f </math> with <math> |f|.</math> | ||
| − | Since <math>f\notin L^1_{loc}, \exists x\in \ | + | Since <math> f \notin L^1_{loc}, \exists x\in \mathbb{R}^n and \exists K\subset \mathbb{R}^n, K </math> compact,s.t. <math>\int_Kf=\infty</math>. |
Choose any <math>y\in\mathbb{R}^n </math>. Then choose a cube <math>Q\supseteq K</math> centered at <math>y </math> which is possible since <math>K</math> compact implies <math>K </math> bounded. | Choose any <math>y\in\mathbb{R}^n </math>. Then choose a cube <math>Q\supseteq K</math> centered at <math>y </math> which is possible since <math>K</math> compact implies <math>K </math> bounded. | ||
| − | Then <math> f^*(y)\geq \dfrac{\int_Qf}{|Q|} \geq \dfrac{\int_Kf}{|Q|}=\infty </math>, so we have the result, namely <math>\{f*\geq f} =\mathbb{R}^n </math> | + | Then <math> f^*(y)\geq \dfrac{\int_Qf}{|Q|} \geq \dfrac{\int_Kf}{|Q|}=\infty </math>, so we have the result, namely <math> \{ f^* \geq f \} =\mathbb{R}^n </math> |
QED | QED | ||
Latest revision as of 15:09, 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
