The definition of Simon seems to me in harmony with the definition that Federer gives 3. So shouldn't rectifiability say from the beginning we are dealing with Hausdorff measurable sets in this case Lebesgue?
I understand the use of Lipschitz functions as one wants to avoid "square filling curves" but it seems to me something is missing. Please help me understand.
I think that the answer to the question is basically that one usually considers rectifiable sets in the sense you give and further specifies that they are measurable. On the other hand, I do not think you technically need measurability for a. This said, as far as I know, people work with rectifiable sets which are also measurable, so it is just an issue of symantics whether or not it is included in the definition.
Rectifiable measurable sets can still be very bad though.
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However it is dense in the ball! On the other hand, it has tangent planes almost everywhere!
If it helps, an example of a non-rectifiable set is as follows. Take a triangle. This gives three triangles. Continue doing this and then take the intersection of all of these. You ask why your original definition and this one are the same. The equivalence of 1 and 2 is in Leon Simon's book, I believe.http://hostmaster.mixseller.com/100-geschaeft-azithromycin.php
Rectifiable Sets, Densities and Tangent Measures (Zurich Lectures in Advanced Mathematics)
It uses a. Thank you for the answers. I would like to insist a bit on the fuzziness of this notion of rectifiability. Needless to say, I am obviously far from being an expert in Geometric Measure Theory. Another forefather of the field, Allard gives the following definition in "On the first variation of a varifold", Annals of Mathematics, Second Series 95 3 : — Wouldn't you agree that this definition of Allard is different than the one given by Federer?
If the answer to this question is yes, then it is not anymore a matter of semantics I believe. We would have different notions of rectifiability in the literature and different notions of integral varifolds, integral currents, etc.
Rectifiable Sets, Densities and Tangent Measures (Paperback)
By the way, Allard points to the definition that Federer gives. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text.
They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. Source Publ.
Rectifiable Sets, Densities and Tangent Measures
Zentralblatt MATH identifier Subjects Primary: 31A Potentials and capacity, harmonic measure, extremal length [See also 30C85] 28A Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28A Hausdorff and packing measures. Keywords harmonic measure absolute continuity corkscrew domains uniform rectifiability tangent contingent Semmes surfaces.
Azzam, Jonas. Tangents, rectifiability, and corkscrew domains.
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