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Let \(\mathcal {I}_0\) be a compact interval in \( \mathbb {R}^m\) (or \(\mathbb {R}^1\) ) and \( \mathcal {E} \subset \mathbb {R}^m\) (or \(\mathbb {R}\) ) a measurable subset. Also, \(\mu _\infty ({\mathcal {E}})\) stands for the Lebesgue measure of \(\mathcal {E}\) . The Lebesgue integral of a function \( g\) over the set \( \mathcal {E} \) will be denoted by \( L\int \limits _{\mathcal {E}} g.\) Throughout the chapter, \( \mathcal {X}\) is a real Banach space with norm \(||.|| \) and \(\mathcal {X}^* \) is its dual. \( B_{\mathcal {X}}^*= \{ y^* \in \mathcal {X} : ||y^*|| \leq 1 \} \) is the closed unit ball in \( \mathcal {X}^*\) .

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Countable Additivity of Henstock-Dunford Integrable Function and Orlicz Spaces

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  • 发表日期：

    2025年

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  • Bipan Hazarika,
  • Hemanta Kalita

摘要

Let \(\mathcal {I}_0\) be a compact interval in \( \mathbb {R}^m\) (or \(\mathbb {R}^1\) ) and \( \mathcal {E} \subset \mathbb {R}^m\) (or \(\mathbb {R}\) ) a measurable subset. Also, \(\mu _\infty ({\mathcal {E}})\) stands for the Lebesgue measure of \(\mathcal {E}\) . The Lebesgue integral of a function \( g\) over the set \( \mathcal {E} \) will be denoted by \( L\int \limits _{\mathcal {E}} g.\) Throughout the chapter, \( \mathcal {X}\) is a real Banach space with norm \(||.|| \) and \(\mathcal {X}^* \) is its dual. \( B_{\mathcal {X}}^*= \{ y^* \in \mathcal {X} : ||y^*|| \leq 1 \} \) is the closed unit ball in \( \mathcal {X}^*\) .