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This chapter extends the concepts of Lebesgue measure and integration to higher dimensions. The structure of open sets in \(\mathbb {R}^2\) is studied. Topics include Lebesgue outer measure, measure, measurable functions, and Lebesgue integration in \(\mathbb {R}^N\) . Additionally, Fubini’s theorem and its consequences are explored, providing a deeper understanding of measure and integration in higher-dimensional spaces.

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Lebesgue Measure and Integration in \(\mathbb {R}^N\)

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  • 发表日期：

    2025年

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  • Satya N. Mukhopadhyay,
  • Subhasis Ray

摘要

This chapter extends the concepts of Lebesgue measure and integration to higher dimensions. The structure of open sets in \(\mathbb {R}^2\) is studied. Topics include Lebesgue outer measure, measure, measurable functions, and Lebesgue integration in \(\mathbb {R}^N\) . Additionally, Fubini’s theorem and its consequences are explored, providing a deeper understanding of measure and integration in higher-dimensional spaces.