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non-measurable functions; rational automorphism
This note contains a proof of the existence of a one-to-one function $\Theta $ of $\,\mathbb{R}\,$ onto itself with the following properties: $\Theta $ is a rational-linear automorphism of $\mathbb{R}$, and the graph of $\Theta $ is a non-measurable subset of the plane.
[1] Gelbaum B. R., Olmsted J. M. H.: Counterexamples in Analysis. The Mathesis Series Holden-Day, San Francisco, 1964. MR 0169961
[2] Kharazishvili A. B.: Nonmeasurable Sets and Functions. North-Holland Mathematics Studies, 195, Elsevier Science B.V., Amsterdam, 2004. MR 2067444
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