# Talk:Rectifiable set

From Encyclopedia of Mathematics

**Definition 2**: is it clear what is meant by "$k$-dimensional graphs of $\mathbb R^n$"? --Boris Tsirelson 09:43, 4 August 2012 (CEST)

As far as I remember, Hausdorff dimension *k* does not imply finite (or even σ-finite) *k*-dimensional Hausdorff measure. Really so? And then "might not be $\mathcal{H}^k$-measurable" looks strange. Is it OK? Do the definitions implicit here conform to definitions in Hausdorff measure and Hausdorff dimension? --Boris Tsirelson 09:50, 4 August 2012 (CEST)

- Sorry, I was saving while constructing the page... I guess I should use a Sandbox instead, sorry again for wasting your time. I have to check still everything once again, since I just typed. Anyway:

- For Lipschitz graphs I can add an explanation if you think it is needed.
- For consistence with the pages Hausdorff measure and Hausdorff dimension I will check later (it is in my "to do list").
- You are right: Hausdorff dimension k does not imply finite (or even σ-finite). If you are referring to the decomposition of a general set in rectifiable and purely unrectifiable part you need the $\sigma$-finiteness hypothesis (which I just added). Rectifiable sets are automatically $\sigma$-finite because of the covering with countably many $C^1$ submanifolds (a $C^1$ submanifold is automatically $\sigma$-finite). The $\sigma$-finiteness assumption might be needed somewhere else and it is the typical thing on which I might slip inadvertently: I will check again everything with care.
- If I drop the Borel assumption in the definition of rectifiability, you might have the following example: Take a common $C^1$ injective curve $\gamma: [0,1]\to\mathbb R^2$ and take the typical Vitali non-Lebesgue measurable subset $V\subset [0,1]$. Now $\gamma (V)$ has Hausdorff dimension $1$ and it can be covered by a single one-dimensional submanifold, so it is
*rectifiable without Borel assumption*. But it is not $\mathcal{H}^1$ measurable. Do you think it is worth to add this example?

Camillo 10:15, 4 August 2012 (CEST)

- I see. Yes, sometimes sandbox is worth to use. Well, now I wonder, why Hausdorff dimension
*k*is stipulated in these three definitions. It follows from the other requirements that it cannot exceed*k*, right? Thus, you just want to exclude the degenerate case, the dimension less than*k*, right? But then, did you exclude the case of dimension*k*but zero $\mathcal{H}^k$? --Boris Tsirelson 13:14, 4 August 2012 (CEST)

- I see. Yes, sometimes sandbox is worth to use. Well, now I wonder, why Hausdorff dimension

- Well, this is a notational problem on which there is no unique convention. As you point out, I specified the dimension because otherwise a $1$-dimensional curve or a fractal of dimension $3/2$ might also be considered as a $2$-dimensional rectifiable set and this is kind of awkward... But I prefer to include $k$-dimensional sets which are $\mathcal{H}^k$-null sets in the class of rectifiable ones. The way it is stated now respects this convention. Camillo 14:04, 4 August 2012 (CEST)

- I see: you know what you're doing. But I bother that another reader may also (similarly to me) got puzzled, thinking that your intention is to bound the dimension from above. A clarifying phrase could help. --Boris Tsirelson 16:23, 4 August 2012 (CEST)

- Indeed. I added a remark. I also mentioned something more about non $\sigma$-finite sets. Camillo 23:08, 4 August 2012 (CEST)

- I also turned "Besicovitch-Federer projection theorem" into a redirect page. Feel free to change or revert if you prefer another form of reference. --Boris Tsirelson 13:22, 4 August 2012 (CEST)
- Definitely a cleaner solution: I will do it also with the other pages I just created.Camillo 14:04, 4 August 2012 (CEST)

- I also turned "Besicovitch-Federer projection theorem" into a redirect page. Feel free to change or revert if you prefer another form of reference. --Boris Tsirelson 13:22, 4 August 2012 (CEST)

**How to Cite This Entry:**

Rectifiable set.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Rectifiable_set&oldid=27377