习题 19, § 5

令 $\mu$ 为$T$ 的正测度。 a) 令 $(f_n)$ 为 $\mathscr{L}^p$ ($1 < p < +\infty$) 中函数的序列，以 $\mathscr{L}^p$ 为界，并收敛于 0。 Show that the sequence $(\widetilde{f}_n)$ converges to 0 in $L^p$ for the weakened topology. b) Give an example of a sequence $(f_n)$ of functions in $\mathscr{L}^1$, bounded in $\mathscr{L}^1$ and convergent in measure to 0, but such that the sequence $(\widetilde{f}_n)$ does not converge to 0 for the weakened topology of $L^1$ (take $\mu$ to be Lebesgue measure on $T = [0,1]$, and use criterion $\alpha$) of Exer. 15). c) Let $\mu$ be Lebesgue measure on $T = [0,1]$. 证明，如果 $f_n(t) = \sin nt$、序列 $(\widetilde{f}_n)$ 对于所有 $L^p$ 的弱化拓扑收敛到 0，使得 $1 \leq p < +\infty$，但 $f_n$ 不收敛到 0。