N維歐氏空間上的均勻分布測度及其切測度

Abstract

這篇論文的重點是研究在R^n上的均勻分布測度以及其切測度的各種幾何性質， 並且儘可能地對這些性質給予仔細而嚴密的證明。這些性質可以用來證明下面的定理： 若 (\Phi) 為 R^n 上的非零測度， m 為小於或等於 n 的非負整數， 且滿足對於所有的 x\in\spt\Phi，和 r>0，皆有 \Phi(B(x,r))=\alpha(m)*r^m 。 如果 m=0,1,2,n，則 (\Phi) 必是支集在 R^n 中某 m 維子空間上的一個 m 維 Hausdorff 測度。 本文中大部份的結果，請參看 David Preiss 的"Geometry of Measures in R^n: Distribution, Rectifiability, and Densities" (Ann. of Math. 125(1987), 537-643.)。

In this thesis we study the uniformly distributed measures in R^n . Some basic properties of these measures and their tangent cones are proved in details. We also give an application of these properties in the last section: Let $\Phi$ be a nonzero measure over $\brn$, and $m\le n$ be an integer so that $\Phi(B(x,r))=\alpha(m)*r^{m}$, for every $x\in\spt\Phi$ and every $r>0$. In case of $m=0,1,2,n$, there is an $m$ affine subspace $V$ of $\brn$ such that $\Phi=H^m| V$. Most of the results in this thesis may be found in David Preiss' paper ([P]).