Functions of Baire class one

Abstract

Let K be a compact metric space. A real-valued function on K is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. We study two well known ordinal indices of Baire-1 functions, the oscillation index β and the convergence index γ. It is shown that these two indices are fully compatible in the following sense: a Baire-1 function f satisfies β(f) ≤ ω ξ1 · ω ξ2 for some countable ordinals ξ 1 and ξ 2 if and only if there exists a sequence (f n) of Baire-1 functions converging to f pointwise such that sup n β(f n) ≤ ω ξ1 and γ((f n)) ≤ ω ξ2. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if α(f) ≤ ω ξ1 and β(g) ≤ ω ξ2, then β(fg) ≤ ω ξ, where ξ = max{ξ 1 + ξ 2, ξ 2 + ξ 1}. These results do not assume the boundedness of the functions involved.