It is well known that boundedness of a subadditive function need not imply its continuity. Here we prove that each subadditive function $$f:X\rightarrow {\mathbb {R}}$$ f : X → R bounded above on a shift–compact (non–Haar–null, non–Haar–meagre) set is locally bounded at each point of the domain. Our results refer to results from Kuczma’s book (An Introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, 2nd edn, Birkhäuser Verlag, Basel, 2009, Chapter 16) and papers by Bingham and Ostaszewski [Proc Am Math Soc 136(12):4257–4266, 2008, Aequationes Math 78(3):257–270, 2009, Dissert Math 472:138pp., 2010, Indag Math (N.S.) 29:687–713, 2018, Aequationes Math 93(2):351–369, 2019).

中文翻译：

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关于在“大”集合上有界的次可加函数

众所周知，次可加函数的有界性并不意味着它的连续性。这里我们证明了每个子可加函数 $$f:X\rightarrow {\mathbb {R}}$$ f : X → R 上界在一个移位紧凑（非-Haar-null，非-Haar-meagre）集合上是在域的每个点局部有界。我们的结果参考了 Kuczma 的书（函数方程和不等式理论介绍。Cauchy 方程和 Jensen 不等式，第 2 版，Birkhäuser Verlag，巴塞尔，2009 年，第 16 章）以及 Bingham 和 Ostaszewski 的论文 [Proc Am Math Soc 136(12):4257–4266, 2008, Aequations Math 78(3):257–270, 2009, Dissert Math 472:138pp., 2010, Indag Math (NS) 29:687–713, 2018, equations Math(2018) 2):351–369, 2019)。