Transformations into Baire 1 functions

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 ? h is equivalent to a Baire I 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 ? h is not measurable. It is shown here that if f ? h is measurable, for every homeomorphism h, then there is an h for which f o h is equivalent to a Baire I function. 1. 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 [1] has obtained the following interesting contrasting facts. If f assumes finitely many values there is a homeomorphism h of I such that f o 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 functionf o h is not measurable. A function f is said to be absolutely measurable if for every homeomorphism h of I the function f o h is measurable. This is tantamount to saying thatf is measurable with respect to every Lebesgue-Stieltjes measure derived from a strictly increasing continuous distribution function. We prove the following result. THEOREM. If f: I -* R is absolutely measurable there is a homeomorphism h of I such that f o h is equivalent to a Baire 1 function. 2. We give some preliminary definitions and lemmas. A set E C I is of absolute measure zero if for every homeomorphism h the set h(E) is of measure zero. A point x E I is a c-point of a set E if for every neighborhood N of x the set N n E has cardinality c and x is a perfect c-point of E if for every neighborhood N of x the set N n E contains a nonempty perfect set. E is c-dense (perfectly dense) in a set D if every point of D is a c-point (perfect c-point) of E. Gorman [2] has obtained the following lemma. LEMMA 1. If E c I is of the first category there is a homeomorphism h of I Received by the editors March 18, 1977. AMS (MOS) subject classifications (1970). Primary 26A21. 'Supported in part by NSF Grant No. MCS-76-06573. ? American Mathematical Society 1977