A Basic Introduction of Geometric Measure Theory

Geometric measure theory studies properties of measures, functions and sets. In this note, we provide a basic introduction of geometric measure theory. We will study elementary properties of Hausdorff measures, Lipschitz functions and countably rectifiable sets. To make this note easily accessible, we confine our discussions in Euclidean spaces. Important topics include the area and coarea formulae for Lipschitz functions and approximate tangent space properties of countably rectifiable sets. This note is prepared for lectures in the Special Lecture Series the author delivered in Peking University, June 2006. He would like to thank Professor Gang Tian for providing such an opportunity. He would also like to thank Miss Xianghui Yu for helping him dealing various issues when he was visiting Peking University. Last but not least, the author would like to thank all participants of this series of lectures. Because of their active participation in class and numerous discussions with him after the class, the author was able to correct many mistakes in the early version of this lecture note.