Definition 4

Given a clan $\Phi$ of subsets of a set $A$, and a Banach space $F$, one calls step function² over the sets of $\Phi$ (or $\Phi$-step function), with values in $F$, every function of the form $\sum_i a_i \varphi_{M_i}$, where the $a_i$ belong to $F$, and the $M_i$ to $\Phi$.

It is clear that the set $\mathscr{E}_F(\Phi)$ of $\Phi$-step functions with values in $F$ is a vector space over $\mathbf{R}$ or $\mathbf{C}$. We have just seen in Prop. 17 that the set $\mathscr{E}(\Phi)$ of real-valued $\Phi$-step functions is an algebra over $\mathbf{R}$; it is also the linear subspace of $\mathbf{R}^A$ generated by the characteristic functions of the sets of $\Phi$.

By the Lemma, every function in $\mathscr{E}_F(\Phi)$ may be written $f = \sum_j c_j \varphi_{N_j}$, where the $N_j \in \Phi$ are pairwise disjoint; from this it follows that $|f| = \sum_j |c_j| \varphi_{N_j}$ belongs to $\mathscr{E}(\Phi)$. In particular, $\mathscr{E}(\Phi)$ is a Riesz space, since the upper envelope of two functions in $\mathscr{E}(\Phi)$ belongs to $\mathscr{E}(\Phi)$.