Transformations into Baire $1$ functions

Abstract:

A measurable f from $I = [0,1]$ to R is equivalent to a Baire 2 function but may not be equivalent to any Baire 1 function. Gorman has obtained the following interesting contrasting facts. If f assumes finitely many values there is a homeomorphism h of I such that $f \circ h$ is equivalent to a Baire 1 function, but there is a measurable f which assumes countably many values which does not have this property. However, the example of Gorman is such that for some homeomorphisms h the function $f \circ h$ is not measurable. It is shown here that if $f \circ h$ is measurable, for every homeomorphism h, then there is an h for which $f \circ h$ is equivalent to a Baire 1 function.