Sets on which measurable functions are determined by their range


Maxim R.Burke & Krzysztof Ciesielski

Canad. J. Math. 49 (1997) 1089-1116.

We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.

In a paper [BD: A. Berarducci, D. Dikranjan, Uniformly approachable functions and UA spaces, Rend. Ist. Matematico Univ. di Trieste 25 (1993), 23--56] it is shown that, under the Continuum Hypothesis (CH), in any separable Baire topological space X there is a set M such that for any two continuous real-valued functions f and g on X, if f and g are not constant on any nonvoid open set and f[M] is a subset of g[M] then f=g. In particular, if f[M]=g[M] then f=g. Sets having this last property with respect to entire (analytic) functions in the complex plane were studied in [DPR: H.G. Diamond, C. Pomerance, L. Rubel, Sets on which an entire function is determined by its range, Math Z. 176 (1981), 383--398] where they were called sets of range uniqueness (SRU's). We study the properties of such sets in measurable spaces with negligibles. We prove a generalization of the aforementioned result from [BD] to such spaces and answer Question 1 from [BD] by showing that CH cannot be omitted from the hypothesis of their theorem. We also study the descriptive nature of SRU's for the nowhere constant continuous functions on Baire Tychonoff topological spaces.

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Last modified October 20, 2001.