$\lim_{n\to\infty}\left\Vert f_n-f\right\Vert _{L^1}=0, \mbox{且} f_n \to f$?

flyy

设 $\displaystyle \left\{f_n\right\}\subset L^1(\mathbb{R}^d)$, 且

$$\begin{aligned} \exists\ p\gt 1,\mathrm{ s.t.} \left\Vert f_{n+1}-f_n\right\Vert _{L^1}\leq\frac{1}{n^{2p}}, \forall\ n\geq 1. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

试证明:

$$\begin{aligned} \exists\ f\in L^1(\mathbb{R}^d),\mathrm{ s.t.} \lim_{n\to\infty}\left\Vert f_n-f\right\Vert _{L^1}=0, \mbox{且} f_n \to f, \mbox{a.e.} x\in\mathbb{R}^d. \tiny\boxed{\begin{array}{c}\mbox{跟锦数学微信公众号}\\\mbox{zhangzujin.cn}\end{array}}\end{aligned}$$

zhangzujin

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