Exercise 9, § 5

Let $\mu$ and $\nu$ be two alien positive measures on $T$. Show that, for every number $p$ such that $1 \leq p \leq +\infty$, the topological vector space $L^p(T, \mu + \nu)$ is isomorphic to the product space of the topological vector spaces $L^p(T, \mu)$ and $L^p(T, \nu)$.