Exercise 20, § 5

a) Let $(f_n)$ be a sequence of functions in $\mathscr{L}^1$ such that: 1° the sequence $(f_n)$ converges in measure to 0; 2° for every $\varepsilon > 0$, there exists an $\eta > 0$ such that the relation $\mu^*(A) \leq \eta$ implies $\limsup_{n \to \infty} \int_A^* |f_n| d\mu \leq \varepsilon$; 3° for every $\varepsilon > 0$, there exists an integrable set $B$ such that $\int_{T - B} |f_n| d\mu \leq \varepsilon$ for all $n$. Show that the sequence $(f_n)$ converges in mean to 0. b) Let $\mu$ be Lebesgue measure on $T = [0,1]$. For $1 < p < +\infty$, denote by $f_n$ the function equal to $n^{2/p}$ in the interval $\left[ \frac{1}{n+1}, \frac{1}{n} \right]$, and to 0 elsewhere. Show that the sequence $(f_n)$ satisfies the three conditions of a) and that the sequence $(\widetilde{f}_n)$ converges to 0 for the weakened topology of $L^p$, but not for the topology of convergence in mean of order $p$.