Extension property for equi-Lebesgue families of functions

Abstract

Let $X$ be a topological space and $(Y,d)$ be a complete separable metric space. For a family $\mathscr F$ of functions from $X$ to $Y$ we say that $\mathscr F$ is equi-Lebesgue if for every $\varepsilon >0$ there is a cover $(F_n)$ of $X$ consisting of closed sets such that ${\rm diam\,}f(F_n)\leq \varepsilon$ for all $n\in\mathbb N$ and $f\in\mathscr F$.

We prove that if $X$ is a perfectly normal space, $Y$ is a complete separable metric space and $E\subseteq X$ is an arbitrary set, then every equi-continuous family $\mathscr F\subseteq Y^E$ can be extended to an equi-Lebesgue family $\mathscr G\subseteq Y^X$.