Corollary

If $\varphi$ is a mapping of $\prod_n F_n$ into a set G, then the mappings $\varphi((f_n))$ and $\varphi((g_n))$ of X into G are equivalent.

We denote by $\varphi((\tilde{f}_n))$ the equivalence class of every function $\varphi((f_n))$, where $f_n$ is an arbitrary function in the class $\tilde{f}_n$.

In particular, if F is a vector space over $\mathbf{R}$, one defines $\tilde{f} + \tilde{g}$ and $\alpha \tilde{f}$ to be the equivalence classes of $f + g$ and $\alpha f$, respectively ($f$ and $g$ being mappings of X into F, and $\alpha$ a scalar); we obtain in this way, on the set of equivalence classes of mappings of X into F, a vector space structure: moreover, this is the quotient space structure of that of $F^X$ by the linear subspace of mappings $f$ such that $\tilde{f} = \tilde{0}$ (the functions that are zero almost everywhere), which we also call negligible functions (with values in F). One defines similarly the product $\tilde{g} \tilde{f}$, where $\tilde{f}$ is an equivalence class of mappings of X into F, and $\tilde{g}$ is an equivalence class of (finite) numerical functions defined on X: the set of equivalence classes of mappings of X into F is thus equipped with the structure of a module over the set of equivalence classes of finite numerical functions defined on X (which is itself equipped with a ring structure). If F is an algebra over $\mathbf{R}$, one defines similarly an algebra structure on the set of equivalence classes of mappings of X into F.

Let F be a metrizable topological space, and consider a uniform structure on F compatible with its topology and defined by a countable family of pseudometrics $\rho_n$ (GT, IX, §§1 and 2); in order that two mappings $f, g$ of X into F be equivalent, it is necessary and sufficient that the numerical functions $\rho_n(f, g)$ be negligible; for, this is equivalent to saying that there exists a negligible set H in X such that, for every $x \notin H$, $\rho_n(f(x), g(x)) = 0$ for all $n$, that is, $f(x) = g(x)$. In particular, if F is a metrizable locally convex space, and $(q_n)$ is a countable family of seminorms defining the topology of F (TVS, II, §4, No. 1), in order that two mappings $f, g$ of X into F be equivalent it is necessary and sufficient that all of the numerical functions $q_n(f(x) - g(x))$ be negligible.