Proposition 2

The set of step functions defined on $I$, with values in a vector space $E$ over $\mathbf{R}$, is a vector subspace $\mathscr{E}$ of the vector space $\mathscr{F}(I; E)$ of all maps of $I$ into $E$.

Indeed, let $f$ and $g$ be two step functions, and $(A_i)$ and $(B_j)$ two partitions of $I$ into a finite number of intervals such that $f$ (resp. $g$) is constant on each of the $A_i$ (resp. $B_j$); whatever the real numbers $\lambda, \mu$, it is clear that $\lambda f + \mu g$ is constant on each of the nonempty intervals $A_i \cap B_j$, and that these intervals form a partition of $I$.