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In Chapter 2 (Example 2C), we considered the Lebesgue measure λ on the Borel σ-algebra ℜ of subsets of the real line ℝ, which is generated by the collection of all open intervals; and we have been using the notion of Lebesgue measure since then, although it has not been properly constructed so far. Indeed, we promised in Example 2C to prove existence and uniqueness of the Lebesgue measure λ: ℜ → \(\overline{\mathbb{R}}\) in Chapter 8. We will comply with that promise in Section 8.3 as a special case of the following program.

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Extension of Measures

  • 章节

  • 发表日期：

    2026年

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  • Carlos S. Kubrusly

摘要

In Chapter 2 (Example 2C), we considered the Lebesgue measure λ on the Borel σ-algebra ℜ of subsets of the real line ℝ, which is generated by the collection of all open intervals; and we have been using the notion of Lebesgue measure since then, although it has not been properly constructed so far. Indeed, we promised in Example 2C to prove existence and uniqueness of the Lebesgue measure λ: ℜ → \(\overline{\mathbb{R}}\) in Chapter 8. We will comply with that promise in Section 8.3 as a special case of the following program.