Wow, thanks!

Thanks! Glad you liked it.

Wonderful! That will do.

Yes. But is the length distance not compact? For example, is it not true that every sequence has a convergent subsequence?

Yes. Lambda is the percentage of "surviving" intervals. We keep increasing them in order to preserve as much as we can. In limit, you remove almost nothing, so lambda is 50% -- remember there are two copies of size lambda. So, in limit, you basically remove nothing. The denominators and log work out to give that inequality.

@@YoungMeasures thanks.

@@YoungMeasures have you made some video on uniform rectifiability? I have been referred to your channel long back by some professors in Finland. Your content is very good. Keep doing this.

@@sushobhandas7752 I have no videos on that particular topic. I would look up Jonas Azzam’s channel for them.

@@YoungMeasures can you share the link of that video or channel here.

Yes yes. I did not know not uniform equicontinuity, i.e., pointwise equicontinuity, was a thing until this comment! LoL!

It is called CARATHEODORY^S CRITERION. It is Theorem 5 in first edition of Measure Theory and Fine Properties of Functions, by C. Evans and F. Gariepy. Other advanced books must have a proof too.

@@YoungMeasures Thank you!

Thanks. Any GMT book will do. Examples? I do not know of one doing examples as explicit. Maybe they toss them as exercises. I did come up with them for the lecture but to be fair, they are easy to come up with.

I have a video on different books for GMT. Might help to check out.

Noted!

The standard book on the topic is: Capogna, Luca and Danielli, Donatella and Pauls, Scott D. and Tyson, Jeremy T., An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. But depending on you need and focus there may be other (possibly non-book) options. If you give more details I might find better references --Young measures

Lectures on Analysis on Metric Spaces by Juha Heinonen Sobolev Spaces on Metric Measure Spaces An Approach based on Upper Gradients Juha Heinonen Pekka Koskela Nageswari Shanmugalingam Jeremy T. Tyson Topics on Analysis in Metric spaces, Luigi Ambrosio, Paolo Tilli. First published in 2004, approximatly 130 pp. Easy to read? Yes. Exercises? Yes. Nice ones! Focuses on: Lipschitz curves/maps into metric spaces, geodesic problem, Sobolev spaces on metric spaces

@@YoungMeasures Thank you very much sir for suggesting these books. Appreciated!

Yes. I think you are right! The least confusing way to state it is to assume both finite and positive. I will either cut it out or add clarification. Thanks.

Unfortunately, I am not aware of any references. But was boundedness used in any essential way?

Sure. Please do.

I don't know why the comment with the link is being deleted ... Could I send it by any other mean?

​@@YoungMeasures I have sent it via your email in the description of the channel

Of course. On channel descriptions you can find my emails.

@@YoungMeasures I sent it to your gmail. Thank you

Thanks for the feedback! I have covered multiple topics, so if you tell me which playlist or topic you like most and also with what math you need to learn, I can find some titles for you.

Around min 23 I explain that we identify G with g via exp map. This then means that in our special setting the exponential map is the identity! It is initially a bit difficult to live with this but you can go over the process to convince yourself that it is true! The real reason is Campbell Hausdorff formula.

Absolutely. Up to precision with a.e. representations, the two classes should coincide. How easy it is to prove depends on how you define the Sobolev functions between manifolds. The Newtonian-Sobolev spaces are also defined for maps into Banack space targets as well. I will refer to the book by Jeremy T. Tyson, Nageswari Shanmugalingam, Juha Heinonen, Pekka Koskela.

I refer to Azzam’s channel for these. He has fantastic videos on GMT. kzbin.info/aero/PLp0TNqYe2DSEA8r9xYJnJCibS1pN-NQq3&si=M7aYBvhygsEaGkwt

I do not understand the question itself. Do you want a proof of the 5r covering lemma in metric setting?

It is a complete metric space yes. For example because convergence with respect to carnot distance is equivalent to convergence as if in usual R^3. Do you mean if there is a norm on H^1 that is, say, bilipschitz to the Koranyi, and as a 3D vector space over R, H^1 becomes a Bsnach space? I have to say no, but dont have a simple readon why. Maybe the scaling would provide a contradiction. Because scaling that is consistent with Carnot geometry is different from the usual scalar product.

@@YoungMeasures That's right, the only thing we don't have for the Koranyi norm is the property ||ax|| = |a|.||x|| and to define Banach space, it is assumed that a norm satisfies such properties. My question is precisely whether I could either define a norm in H^1 that makes it Banach, or whether the definition of Banach space (but maintaining the known properties) can be relaxed to meet this case in which the norm is not homogeneous. Maybe this isn't trivial, or doesn't make sense, as I haven't found anything about it.

Will do soon.

Ok, Simon has some notes online, which is basically a newer version of his book's material. There in Remark 3.4 it is stated that: Fix an (orthogonal) basis of R^(n+m). If projection of a set S onto any of the hyperplanes given by span of n-many of these basis vectors has zero Hausdorff-n measure then S is purely H^n-unrectifiable.

If you know or can find a reference for that fact that in any rectifiable set with positive measure you can find a set with positive measure that is bi-Lip to a subset of Euclidean space with biLip constant as close to 1 as we wish, then this should not be difficult to see: Working locally, the projection of a part of the set onto a suitable tangent space (a hyperplane) must have positive measure. Because the set and its projection onto the tangent are (working on even smaller neighborhoods) extremely nearby, the projection of the set to the hyperplanes that are in general position with regard to that tangent plane, is pretty much the same projecting the set onto tangent plane first then projecting onto that hyperplane. Since projection onto the hyperplane does not vanish the area, we are left with a projection of S of positive area. (If you want to draw a picture, first draw a 2D manifold'ish set in 3D, i.e. S, then a tanget plane to it, the tangent plane projects nontrivially to one of xy, or xz or yz planes. )

Thank you for very clear references.