Exercise 19, § 5

Let $\mu$ be a positive measure on $T$. a) Let $(f_n)$ be a sequence of functions in $\mathscr{L}^p$ ($1 < p < +\infty$) bounded in $\mathscr{L}^p$ and convergent in measure to 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 $T = [0,1]$. Show that, if $f_n(t) = \sin nt$, the sequence $(\widetilde{f}_n)$ converges to 0 for the weakened topology of all the $L^p$ such that $1 \leq p < +\infty$, but $f_n$ does not converge in measure to 0.