Corollary 3

Let $f$ be a moderated function defined on T. There exists a sequence $(K_n)$ of pairwise disjoint compact sets, and a negligible set H, such that $f = f \varphi_H + \sum_n f \varphi_{K_n}$.

For, let G be a set that is a countable union of integrable open sets, such that $f$ is zero outside G; then G is the union of a sequence $(K_n)$ of pairwise disjoint compact sets and a locally negligible set H (No. 8, Prop. 11); but H is moderated, therefore negligible.