Exercise 15, § 5

Let H be a bounded subset of the space $\mathscr{L}^1(T, \mu)$, $\widetilde{H}$ its canonical image in the Banach space $L^1(T, \mu)$. Show that the following conditions are equivalent:

$\alpha$ The following set of two conditions holds: $\alpha_1$ for every $\varepsilon > 0$, there exists a compact set $L \subset T$ such that $\int_{T - L} |f| \, d\mu \leq \varepsilon$ for all $f \in H$; $\alpha_2$ for every $\varepsilon > 0$, there exists an $\eta > 0$ such that, for every open set U of T such that $\mu^*(U) \leq \eta$, one has $\int_U |f| \, d\mu \leq \varepsilon$ for all $f \in H$.

$\beta$ Both condition $\alpha_1$ and the following condition $\beta_2$ hold: $\beta_2$ for every compact set $K \subset T$ and every $\varepsilon > 0$, there exists an open neighborhood U of K such that $\int_{U - K} |f| \, d\mu \leq \varepsilon$ for all $f \in H$.

γ) For every sequence $(g_n)$ of functions in $\mathscr{L}^\infty$, uniformly bounded and convergent in measure to a function $g$, one has $\lim_{n \to \infty} \int f g_n \, d\mu = \int f g \, d\mu$ uniformly as $f$ runs over $\mathrm{H}$.

δ) For every sequence $(h_n)$ of functions continuous on $\mathrm{T}$ and tending to 0 at the point at infinity, uniformly bounded and such that $\lim_{n \to \infty} h_n(t) = 0$ for every $t \in \mathrm{T}$, one has $\lim_{n \to \infty} \int f h_n \, d\mu = 0$ uniformly as $f$ runs over $\mathrm{H}$.

ζ) For every infinite sequence $(\mathrm{U}_n)$ of pairwise disjoint open sets of $\mathrm{T}$, one has $\lim_{n \to \infty} \int_{\mathrm{U}_n} f \, d\mu = 0$ uniformly as $f$ runs over $\mathrm{H}$.

(To show that ζ) implies β) and that β) implies α), use a 'gliding hump method' in the same way as in Exer. 13 b). Then prove that α) implies γ), that γ) implies δ), and that δ) implies ζ).)

If it is assumed moreover that, for every $t \in \mathrm{T}$ such that $\mu(\{t\}) \neq 0$, the set of $f(t)$ is bounded in $\mathbf{R}$ as $f$ runs over $\mathrm{H}$, show that the preceding conditions are equivalent to the condition:

θ) $\widetilde{\mathrm{H}}$ is a relatively compact subset of $L^1$ for the weakened topology $\sigma(L^1, L^\infty)$.

(To show that θ) implies β), argue by contradiction, using Šmulian's theorem (TVS, IV, §5, No. 3, Th. 2) and a 'gliding hump method'. To show that α) implies θ), show that $\mathrm{H}$ is bounded in $\mathscr{L}^1$, then apply Eberlein's theorem (TVS, IV, §5, No. 3, Th. 1), and make use of Exer. 14.)