Exercise 26, § 1

Let $(f_n)$ be a sequence of $\mu$-integrable functions defined on T, such that for almost every $t \in T$, $\sup_n |f_n(t)| < +\infty$ and $\sup_n \int |f_n(t)|\, d\mu(t) = +\infty$. Show that there exists a sequence $(c_n)$ of scalars such that $\sum_n |c_n| < +\infty$ and such that $\sum_n c_n f_n$ is not $\mu$-integrable (apply the Gelfand–Dunford theorem to the space $L^1(\mathbf{N}) = F$.)