Given a set $A \subset \mathbb{R}^n$, this is called d-rectifiable if it can be covered by a countable union of images of lipshitz functions from $\mathbb{R}^d $ to $ \mathbb{R}^n $ and a $\mathcal{H}^d$ null set. So given $A\_j = f\_j (\mathbb{R}^d)$ with $f\_j$ Lipshitz, one has that A is d-rectifiable if


$$ A \subset \cup\_{j=1} ^\infty A\_j \cup A\_0 \text{ where } \mathcal{H}^d (A\_0)=0 .$$


Now, consider $d=1$ and $n=2$ for simplicity. If I consider a countable number of "bad points", I wouldn't really need that $A\_0$ in the definition. The only relevant case is when the cardinality of $A\_0$ is more than countable.


Now the question is the following. I know there is the structure theorem by Federer-Fleming which tells you that you can decompose any $\mathcal{H}^d $- measurable set of finite measure in a rectifiable part and in a purely unrectifiable part. Let's call this set $A$, the rectifiable part $A^{rect}$ and the unrectifiable part $A^{unr}$. Given the definition above, the only relevant case is when $\mathcal{H}^d (A^{unr}) >0$, otherwise one could consider it to be the null set in the definition, making $A$ rectifiable.


The question is: why did we choose to define rectifiability that way and not without $A\_0$? I mean, one could take $A^{rect}$ to be the union of the $A\_j$s above, while discarding the $A\_0$ in the unrectifiable part. It somehow seems to even make more sense, you could partition a set $A$ in a good part which is ALL contained in unions of images of Lipshitz functions, and in a bad part which contains all the rest. Is this just a convention or there's some good reason to have this definition? I guess it has to do with the fact of wanting to have some "compactness" in the class of rectifiable sets, meaning that a perturbation by a null set don't make you go out of the class, but maybe there's a deeper reason.



Why is there a $\mathcal{H}^d$-null set in the definition of d-rectifiable set?
If $f:\mathbb{R}^n\to\mathbb{R}^m$, $m<n$ is Lipschitz continuous, then for almost all $x\in\mathbb{R}^m$, the set $f^{-1}(x)$ is ($n-m$)-rectifiable according to the definition that includes the null set $A\_0$. (Such sets are often called countably $(n-m)$-rectifiable). This is a nice generalization of the Sard theorem for Lipschitz mappigns. This is one of the main reasons why the definition is stated the way it is stated. When you remove the set $A\_0$ from the definition you will have much less natural and generic examples of rectifiable sets.


