On Representations of Baire One Functions as the Sum of Lower and Upper Semicontinuous Functions

Abstract

According to the Vitali-Carathéodory theorem, the integral of a finite summable function \(f\) on a measurable set may be approximated by the integral of a sum of lower and upper semicontinuous functions. In the case, that \(f\) is a Baire one function, we give the answer to the following question: is there a lower semicontinuous function \(l\) and a upper semicontinuous function \(u\) such that \(f=l+u\) almost everywhere? The answer is in general negative.