Best textbook for Geometric Measure Theory I was wondering what is the best textbook for Geometric Measure Theory for self study. I am looking for one that isnt excessively detailed or long either as I found Rana's Introduction to measure theory fairly slow paced and superfluous to my needs.
What are your thoughts about this one http://www.amazon.co.uk/Geometric-Measure-Theory-Classics-Mathematics/dp/3540606564/ref=sr_1_1?s=books&ie=UTF8&qid=1403868174&sr=1-1&keywords=geometric+measure+theory?
 A: As Dave Renfro pointed out in a comment, Federer's book is famously difficult and advanced - it should probably be avoided as an introduction; I will also second his suggestion of Mattila's Geometry of Sets and Measures in Euclidean Spaces. I find it quite accessible and enjoyable to read; it also has some exercises at the end of each chapter.
Two other books I particularly like are Falconer's Fractal Geometry and Techniques in Fractal Geometry.
A: In this Youtube video, I give a short overview of the most famous ones. I go over contents and talk about what aspects of GMT are emphasized in each. Hope some will find this helpful:
https://www.youtube.com/watch?v=C2NdrGZLGmA
https://drive.google.com/file/d/1bC-duas9lCbd321AlSe-Zy4AGdKCRwcJ/view
A: A good anotated list of textbooks on geometric measure theory can be found in this blog post. Besides comments on Federer and Mattila it has several more examples.
As my personal favorite I found, while lecturing geometric measure theory, "Measure Theory and Fine Properties of Functions" by Evans and Gariepy. It is short and crisp (often you have to build the geometric intuition on your own, but that is ok with me) but the proofs are pretty detailed. Also it contains most things I am interested in. However, for covering theorems like Vitali and Besicovitch, I found Krantz and Parks "Geometric Integration Theory" a bit better organized. Mattila is better suited if you interested in Hausdorff measure and fractals. If you want to know if a certain theorem holds in the case the measure is defined only on a metric space, a Hausdorff topological one or what happens when separability of the underlying space is lost, Bogachev's two volumes on measure theory are the place to look (although, there is not so much "geometry" in there).
