Exercise 9, § 6

Let $X$ be an infinite discrete space, $\mu$ a measure on $X$ such that the support of $\mu$ is equal to $X$. a) Show that for $1 \leq p \leq +\infty$, the space $\mathscr{L}^p(\mu)$ is a topological vector space isomorphic to the space $\mathscr{L}^p(\mu_0)$, where $\mu_0$ is the measure on $X$ defined by the mass +1 at each point of $X$. b) Show that if $1 \leq p < q \leq +\infty$, the topology of convergence in mean of order $p$ is strictly finer than the topology of convergence in mean of order $q$ (on $\mathscr{L}^p$).