Exercise 21, § 5

a) Let $(f_n)$ be a sequence of functions in $\mathscr{L}^1$, such that: 1° $\liminf_{n \to \infty} f_n(t) \geq 0$ almost everywhere in $T$; 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 $n$; 4° $\lim_{n \to \infty} \int f_n d\mu = 0$. Show that the sequence $(f_n)$ converges in mean to 0. (Observe that, for every $\varepsilon > 0$, if $C_m$ is the set of $t \in T$ such that $f_n(t) \geq -\varepsilon$ for at least one $n \geq m$, then the sequence $(C_m)$ is decreasing and has negligible intersection.) b) Let $(f_n)$ be a sequence of functions in $\mathscr{L}^1$ such that: 1° there exists an integrable function $g$ such that $f_n \geq g$ for all $n$; 2° $\liminf_{n \to \infty} f_n(t) \geq 0$ almost everywhere; 3° $\limsup_{n \to \infty} \int f_n d\mu \leq 0$. Show that the sequence $(f_n)$ converges in mean to 0. (First show that, for every measurable set $A$, $\liminf_{n \to \infty} \int_A f_n d\mu \geq 0$; then observe that, for every measurable set $A$,

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