Distances to spaces of Baire one functions

Abstract

Given a metric space X and a Banach space (E, ||·||) we use an index of σ-fragmentability for maps ${f \in E^X}$ to estimate the distance of f to the space B ₁(X, E) of Baire one functions from X into (E, ||·||). When X is Polish we use our estimations for these distances to give a quantitative version of the well known Rosenthal’s result stating that in ${B_1(X, \mathbb{R})}$ the pointwise relatively countably compact sets are pointwise relatively compact. We also obtain a quantitative version of a Srivatsa’s result that states that whenever X is metric any weakly continuous function ${f \in E^X}$ belongs to B ₁(X, E): our result here says that for an arbitrary ${f \in E^X}$ we have

$$d(f, B_1(X, E))\leq 2 \sup_{x^*\in B_{E^{\ast}}}{\rm osc}(x^*\circ f),$$

where osc${(x^{*} \circ f)}$ stands for the supremum of the oscillations of ${x^{*} \circ f}$ at all points ${x \in X}$ . As a consequence of the above we prove that for functions in two variables ${f : X \times K \to \mathbb{R}}$ , X complete metric and K compact, there exists a G _δ-dense set ${D \subset X}$ such that the oscillation of f at each ${(x, k) \in D \times K}$ is bounded by the oscillations of the partial functions f _x and f ^k. A representative result in this direction, that we prove using games, is the following: if X is a σ–β-unfavorable space and K is a compact space, then there exists a dense G _δ-subset D of X such that, for each ${(y, k) \in D\times K}$ ,

$${\rm osc}(f,(y,k))\le 6\sup_{x\in X}{\rm osc}(f_x)+8\sup_{k\in K}{\rm osc}(f^k).$$

When the right hand side of the above inequality is zero we are dealing with separately continuous functions ${f : X \times K \to \mathbb{R}}$ and we obtain as a particular case some well-known results obtained by the third named author in the mid 1970s.