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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?

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    $\begingroup$ Federer's book is rather famous for being quite difficult and advanced. I suggest first considering Geometry of Sets and Measures in Euclidean Spaces by Pertti Mattila. My gut feeling is that anyone who doesn't know anything about Federer's book probably isn't ready to self-study out of Federer's book. (Kind of like the saying "if you have to ask what the price is, you probably can't afford it".) $\endgroup$
    – Dave L. Renfro
    Jun 27, 2014 at 12:13
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    $\begingroup$ I really enjoyed Francesco Maggi's "Sets of Finite Perimeter and Geometric Variational Problems". I'll also second the recommendation for Mattila's book, if you're more into the harmonic analysis side of GMT. $\endgroup$
    – felipeh
    Jun 27, 2014 at 15:23
  • $\begingroup$ @DaveL.Renfro This has just come to my mind.... could you clarify what the difference between geometric measure theory and fractal geometry is please? Its not clear online exactly what the difference is $\endgroup$
    – Trajan
    Jul 8, 2018 at 16:11
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    $\begingroup$ For me, geometric measure theory is when the focus is on things like density properties, intersection and projection properties, and various measure-theoretic properties of the underlying (outer) measures. On the other hand, fractal geometry is when the focus is on things like computing various fractal dimensions and the tools for doing this, such as self-similarity and the open set condition. $\endgroup$
    – Dave L. Renfro
    Jul 8, 2018 at 20:14
  • $\begingroup$ @DaveL.Renfro Is Federer's book really this bad? See .... amazon.com/gp/customer-reviews/R1ED6ATGR34ZXK/…. $\endgroup$
    – Trajan
    Dec 24, 2018 at 18:53

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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).

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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.

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  • $\begingroup$ Ive bought Mattila's book. $\endgroup$
    – Trajan
    Jul 5, 2014 at 9:41
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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

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