Measure of low-dimensional point set in high-dimensional space and integral theory on low-dimensional point set. The establishment of measure theory in the early 20th century made people have a good understanding of the behavior of subsets in Rn about the N-dimensional Lebesgue measure μ n ... Most function theories have undergone great changes because of Lebesgue integral theory. However, difficulties are encountered in dealing with the mathematical problems related to low-dimensional point sets in Rn. For example, the famous Prato problem can be solved by combining the * * shape transformation and Dirichlet principle and skillfully using Lebesgue method in the form of two-dimensional surface. When the dimension of the surface exceeds 2, these classical methods fail. The theory of geometric measure is produced under this background. It begins with 19 14 C Caratheodory's basic work on measure theory. After decades of development, it combines many skills from analysis, geometry and algebraic topology, and produces many new concepts, which has become a powerful tool for mathematical research. During the first 20 to 30 years after Hausdorff measure and integrable set appeared in Caratheodory's work, people's interest was mainly to understand the behavior of subsets in Rn about M-dimensional Hausdorff measure, integral geometric measure and other measures. 0 ≤ k

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Bibliography H. Federer, Geometric Measure Theory, springer Publishing House, Berlin, 1969.