\(\Large\color{red}{\textbf{特征函数}}\textbf{是集列极限与数值极限的桥梁}\)

elim

给定集合\(\Omega,\,\)定义\(A(\subseteq\Omega)\)的特征函数 \({\large\chi}_A:\Omega\to\{0,1\}\)为
\(\qquad{\large\chi}_A(x)=\begin{cases}1,& x\in A,\\0,& x\in\Omega-A.\end{cases}\)
易见 \({\large\chi}_{(\Omega-A)}=1-{\large\chi}_A,\,{\large\chi}_{\bigcup_\alpha A_\alpha}=\sup_\alpha {\large\chi}_{A_\alpha},\,{\large\chi}_{\bigcap_\alpha A_\alpha}=\inf_\alpha{\large\chi}_{A_\alpha}\)
于是 \(\;{\large\chi}_{\overline{\lim} A_n}=\overline{\lim}{\large\chi}_{A_n},\;{\large\chi}_{\underline{\lim} A_n}=\underline{\lim}{\large\chi}_{A_n},\;{\large\chi}_{\lim A_n}=\lim{\large\chi}_{A_n}\)
从特征函数的定义知道 \(x\in A\iff {\large\chi}_A(x) = 1\)故
\(\qquad A=\{x\mid {\large\chi}_A(x)=1\}={\large\chi}_A^{-1}(1)\).

令 \(A_n=\{m\in\mathbb{N}: m>n\}\) 则 \({\large\chi}_{A_n}(x)=\small=\begin{cases}1,& x\ge n+1,\\0,& x< n+1.\end{cases}\)
\(\because\quad\displaystyle{\large\chi}_{\lim A_n}(x)=\lim {\large\chi}_{A_n}(x)=0={\large\chi}_\phi (x)\;(\forall x)\)
\(\therefore\quad{\large\chi}_{\lim A_n}={\large\chi}_\phi,\quad\displaystyle\lim_{n\to\infty}A_n={\large\chi}_{\phi}^{-1}(1)=\phi.\)

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集列极限被相应的特征函数列的极限完全确定，反之亦然。