Equi-Baire 1 families of functions
A family \(\mathcal{F}\) of functions from \(X\) to \(Y\) are said to be equi-Baire 1, if for every \(\varepsilon>0\) there is a function \(\delta_\varepsilon:X\to(0,\infty)\) such that for every \(f\in\mathcal{F}\) and \(x,x'\in X\), the condition \(d_X(x,x')<\min\{\delta_\varepsilon(x),\delta_\varepsilon(x')\}\) implies \(d_Y(f(x),f(x'))<\varepsilon\). This definition is based on a characterization of Baire 1 functions in Polish spaces. The function \(\delta_\varepsilon\) in the definition is called an \(\varepsilon\)-gauge. In the paper the authors study the equi-Baire 1 families of functions between two Polish spaces. They show that the \(\varepsilon\)-gauge for such families can be chosen to be upper-semi-continuous. On the other hand, the lower-semi-continuity of the \(\varepsilon\)-gauge \(\delta_\varepsilon\) sometimes implies the equi-continuity of \(\mathcal{F}\). They prove that a pointwise converging sequence of continuous functions is an equi-Baire 1 family of functions and they characterize equi-Baire 1 families of \(\{0,1\}\)-valued functions. They give an example of a uniformly bounded equi-Baire 1 family of functions on the interval \([0,1]\) that does not converge uniformly. This shows that the Arzelà-Ascoli theorem cannot be generalized to the class of Baire 1 functions. On the other hand they prove the following version of the Arzelà-Ascoli theorem for pointwise convergence: Every pointwise bounded sequence \((f_n)\) of real-valued functions on a separable metric space that forms an equi-Baire 1 family has a subsequence \((f_{k_n})\) pointwise converging to a Baire 1 function.
scientific article; zbMATH DE number 7434160
scientific article