Variations on Lusin’s theorem

Abstract

We prove a theorem about continuous restrictions of Marczewski measurable functions to large sets. This theorem is closely related to the theorem of Lusin about continuous restrictions of Lebesgue measurable functions to sets of positive measure and the theorem of Nikodym and Kuratowski about continuous restrictions of functions with the Baire property (in the wide sense) to residual sets. This theorem is used to establish Lusin-type theorems for universally measurable functions and functions which have the Baire property in the restricted sense. The theorems are shown (under assumption of the Continuum Hypothesis) to be best possible within a certain context. 1. Measurable functions. The results which appear here were announced at the Ninth Summer Symposium on Real Analysis, held at the University of Louisville during 1985, and appeared in abstract form (without proofs) in [9]. We study theorems about functions from a complete metric space X without isolated points (or else from the unit interval I= [0,1]) into the reals R. d will denote the metric for the space X. c denotes the cardinality of the continuum, and CH refers to the Continuum Hypotheses. A perfect set is a closed set M such that every point of M is a limit point of M, and a Cantor set is a homeomorphic image of the middle thirds Cantor subset of I. The measurable functions we will be interested in are defined in terms of the following a-algebras of subsets of X: Bw Baire property in the wide sense [22], Br: Baire property in the restricted sense [22], L: Lebesgue measurable sets (assuming X = I), U: Universally measurable sets (a set M is universally measurable if it is measurable with respect to the completion of every Borel measure on X), (s): Marczewski measurable sets (a set M is Marczewski measurable provided that for every perfect subset P of X, there exists a perfect subset Q of P which either misses M or is a subset of M), and B: Borel measurable. Received by the editors December 26, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 26A15, 28A20; Secondary 54C30, 04A15 .