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On lineability of families of non-measurable functions of two variable

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A function \(F:\mathbb {R}^2\rightarrow \mathbb {R}\) is sup-measurable if, for each (Lebesgue) measurable function \(f:\mathbb {R}\rightarrow \mathbb {R}\), the Carathéodory superposition \(F_f:\mathbb {R}\rightarrow \mathbb {R}\) given by \(F_f: x\mapsto F(x,f(x))\) is measurable. The existence of non-measurable sup-measurable functions is independent of ZFC. We prove, assuming CH, that the family of all non-measurable sup-measurable functions \(F:\mathbb {R}^2\rightarrow \mathbb {R}\) (plus the zero function) contains a linear vector space of dimension \(2^\mathfrak {c}\). A function \(F:\mathbb {R}^2\rightarrow \mathbb {R}\) is separately measurable if all its vertical and horizontal sections are measurable. In the second part of this note we show that the family of non-measurable separately measurable functions is \(2^\mathfrak {c}\)-lineable.

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Acknowledgements

We would like to express our gratitude to Professors Krzysztof C. Ciesielski and Juan B. Seoane–Sepúlveda, and anonymous reviewers for their help in improving this paper’s presentation.

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  1. Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952, Gdańsk, Poland

    Tomasz Natkaniec

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Natkaniec, T. On lineability of families of non-measurable functions of two variable. RACSAM 115, 33 (2021). https://doi.org/10.1007/s13398-020-00980-7

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