The preservation of convergence of measurable functions under composition

Abstract

Let / be a real-valued measurable function on a measure space (S, , p.) and let <p be a Borel measurable function of a real variable.Then it is well-known that the composition <p of, defined by <p of(s) = 4>(f(s)), sES, is also measurable.Conversely, if 4> is not Borel measurable, then there exists a measurable function/ on some measure space such that <f> of is not measurable.We summarize these two remarks in the statement that a function <j> preserves measurability under composition if and only if it is Borel measurable.The purpose of this note is to characterize functions 4> which preserve convergence (in various senses) of measurable functions.After these results were obtained it was pointed out that Theorems 5 and 6 follow from theorems of P. R. Halmos [2].Theorem 7 was proved by M. M. Vanberg [3,, but our proof is much shorter.