In this paper, we formulate necessary and sufficient conditions for relative compactness in the space $$BG({\mathbb {R}}_+,E)$$ of regulated and bounded functions defined on $${\mathbb R}_+$$ with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space $$BG({\mathbb {R}}_+,E)$$ . We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps $$BG({\mathbb {R}}_+,E)$$ into $$BG({\mathbb {R}}_+,E)$$ and, additionally, be compact.

中文翻译：

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无界区间上有规函数空间中的非紧性和叠加算子的测度

在本文中，我们制定了在 $$BG({\mathbb {R}}_+,E)$$ 空间中定义在 $${\mathbb R}_+ 上的调节和有界函数的相对紧致性的充分必要条件$$ 具有 Banach 空间 E 中的值。此外，我们在空间 $$BG({\mathbb {R}}_+,E)$$ 中构造了四个新的非紧致性度量。我们调查了它们的特性，并描述了这些度量之间的关系。我们提供充分必要条件，使得叠加算子 (Niemytskii) 将 $$BG({\mathbb {R}}_+,E)$$ 映射到 $$BG({\mathbb {R}}_+,E) $$，此外，要紧凑。