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In this chapter, we are going to focus on a concept for which the best motivation comes from the relationship between areas (or volumes) and length. From school geometry, we know that the area of a rectangle (resp. volume of a rectangular parallelepiped) equals the product of lengths. Now that we have Lebesgue measure on \(\mathcal{B}({\mathbb R}^1)\) as an extension of length and, at the same time, Lebesgue measure on \(\mathcal{B}({\mathbb R}^k)\) , for any k, as an extension of k-dimensional volume, a natural question to ask would be: is there a connection?

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Product Spaces

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  • 发表日期：

    2025年

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  • Alok Goswami,
  • B. V. Rao

摘要

In this chapter, we are going to focus on a concept for which the best motivation comes from the relationship between areas (or volumes) and length. From school geometry, we know that the area of a rectangle (resp. volume of a rectangular parallelepiped) equals the product of lengths. Now that we have Lebesgue measure on \(\mathcal{B}({\mathbb R}^1)\) as an extension of length and, at the same time, Lebesgue measure on \(\mathcal{B}({\mathbb R}^k)\) , for any k, as an extension of k-dimensional volume, a natural question to ask would be: is there a connection?