Exercise 17, § 5

a) Let $\mu$ be Lebesgue measure on the interval $\mathrm{T} = [0, +\infty[$. Show that if $1 \leqslant r < s < +\infty$, the topologies induced on $L^r \cap L^s$ by the weakened topologies of $L^r$ and $L^s$ are not comparable (cf. Ch. IV, §6, Exer. 8). If $f_n$ is the characteristic function of the interval $[n, n+1]$, show that the sequence $(\widetilde{f}_n)$ tends to 0 for the weakened topology of all the $L^p$ such that $p > 1$, but not for the weakened topology of $L^1$.

b) Let $\lambda$ be Lebesgue measure on the interval $\mathrm{T} = [0, 1]$. Show that if $r < s$, the weakened topology of $L^s$ is strictly finer than the topology on $L^s$ induced by the weakened topology of $L^r$ (cf. Ch. IV, §6, Exer. 8).

c) Let $\mu$ be the positive measure on a discrete space for which every point has measure 1. Show that if $1 \leq r < s < +\infty$, the weakened topology of $L^r$ is strictly finer than the topology induced on $L^r$ by the weakened topology of $L^s$ (Ch. IV, §6, Exer. 9).