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On some properties of Erdős sets. (English) Zbl 0546.39001
Let H be a Hamel basis of \({\mathbb{R}}^ N\). The set \(H^*\) of all finite linear combinations of the elements of H with integral coefficients is called the Erdős set associated with H. Independently of what Hamel basis H we start, the set H always is saturated non-measurable [for the case \(N=1\) see P. Erdős, ibid. 10, 267-269 (1963; Zbl 0123.320)]. The main result of the paper reads as follows. Let \(A\subset {\mathbb{R}}^ N\) be an arbitrary Lebesgue measurable set of positive measure or a second category Baire set, let H be an arbitrary Hamel basis of \({\mathbb{R}}^ N\), and put \(T=A\cap H^*\). If \(D\subset {\mathbb{R}}^ N\) is an open and convex set containing T, and f:\(D\to {\mathbb{R}}\) is a (Jensen) convex function such that f is bounded above on T or the restriction \(f| T\) is continuous, then f is continuous in D.

39B99 Functional equations and inequalities
39B72 Systems of functional equations and inequalities
Zbl 0123.320
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