\(\mathrm {Exercise \ \oplus \ Problem } \ 2 \)
\(\mathrm {Exercise \ \oplus \ Problem } \ 2 \) |
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\( \qquad \)你好，这里是我的个人网站数学分析的每周一题栏目(数学分析每周一题，其中数学分析指的是数学中的分析学, 主要包括微积分，实分析，复分析) \(\qquad \ \)——————Alina Lagrange
Let \(1 \leq p<\infty, L^{p}([0,1], d m)\) be the completion of \(C[0,1]\) with the norm:\( \|f\|_{p}=\left(\int_{0}^{1}|f(x)|^{p} d m\right)^{\frac{1}{p}},\) where \(d m\) is the Lebesgue measure. Show that $$\lim _{\lambda \rightarrow \infty} \lambda^{p} m\{x|| f(x) |>\lambda\}=0$$
\(\mathcal{P}roof. \)
Let \(f\in L^p( 1 \leq p<\infty) \) then $$ \lim _{\lambda \rightarrow \infty} m\left(E= \{x | f(x)>\lambda \} )=0\right . $$ And suppose the set \(e \) satisfies $$m(e)\to 0,e\subset E=[0,1] $$ then $$\int_e |f(x)|\mathrm d m \to 0$$ $$ \int_{0}^{1}|f(x)|^{p} d m=\int_{\{x|| f(x) |>\lambda\}}|f(x)|^{p} d m+\int_{\{x|| f(x) | \leq \lambda\}}|f(x)|^{p} d m $$ then $$ \lambda^{p} m\{x|| f(x) |>\lambda\} \leq \int_{\{x|| f(x) |>\lambda\}}|f(x)|^{p} d m \rightarrow 0, \quad \text { } \lambda \rightarrow \infty $$