Proposition 18

— Let $\lambda$ be an additive set function defined on a clan $\Phi$. There exists one and only one linear form (again denoted $\lambda$) on the vector space $\mathscr{E}(\Phi)$ of real-valued $\Phi$*-step functions*, such that $\lambda(\varphi_M) = \lambda(M)$ for every set $M \in \Phi$; if, moreover, $\lambda(M) \geqslant 0$ for every $M \in \Phi$, then $\lambda$ is a positive linear form on $\mathscr{E}(\Phi)$.

The uniqueness of the linear form $\lambda$ is clear, since the characteristic functions of the sets in $\Phi$ generate the vector space $\mathscr{E}(\Phi)$. To prove the existence of $\lambda$, it suffices to prove that the relation $\sum_i c_i \varphi_{M_i} = 0$, where the $M_i$ are nonempty sets belonging to $\Phi$, implies $\sum_i c_i \lambda(M_i) = 0$.

Now, by the Lemma there exists a finite family $(N_j)$ of pairwise disjoint nonempty sets in $\Phi$ such that, for every index $i$, $\varphi_{M_i} = \sum_j a_{ij} \varphi_{N_j}$ with $a_{ij} = 0$ or $a_{ij} = 1$. The relation $\sum_i c_i \varphi_{M_i} = 0$, which may be written $\sum_j \left( \sum_i c_i a_{ij} \right) \varphi_{N_j} = 0$, therefore implies that $\sum_i c_i a_{ij} = 0$ for every index $j$. By Def. 5, we then have

which proves the existence of $\lambda$. Finally, suppose that $\lambda(M) \geqslant 0$ for every $M \in \Phi$; for every function $f \in \mathscr{E}(\Phi)$, one can write $f = \sum_i c_i \varphi_{M_i}$, where the $M_i \in \Phi$ are pairwise disjoint; if $f \geqslant 0$, it follows that $c_i \geqslant 0$ for every index $i$ such that $M_i$ is nonempty, whence $\lambda(f) = \sum_i c_i \lambda(M_i) \geqslant 0$.