Remark

— It is easily seen that Def. 4 is equivalent to the following: a $\Phi$-step function with values in $F$ is a function $f$ that takes on only a finite number of values and which is such that, for every $a \neq 0$ in $F$, the set $f(a)^{-1}$ belongs to $\Phi$.

² Fonction étagée, whence the notation $\mathscr{E}(\Phi)$ in what follows.