What is the definition of Compact rectifiable set?

Question

In dimension $\mathbb R^N$, we say a set $S$ is $N-1$-rectifiable if there exists a countably many $C^1$ hyper surface $\Gamma_i$ so that $$ \mathcal H^{N-1}(S_u\setminus \bigcup\Gamma_i)=0 $$

Now I am reading a paper it use the term compact rectifiable set without define it, and I can't find a reliable reference where it has the definition of this term. Also, the paper uses 'compact rectifiable set' and 'closed rectifiable set' together, so it looks to me they are the same thing.

Can any body point out a good reference which has the definition of them?

Also, if the set $S$ so that $\mathcal H^{N-1}(S)<\infty$. Then by properties of measure, I can always have a compact set $K\subset S$ so that $\mathcal H^{N-1}(S\setminus K)<\epsilon$ for arbitrary $\epsilon>0$. Then, can I say the set $K$ is compact rectifiable?

Thank you!

Answer 1

Definition from Munkres : Let $S$ be a bounded set in $\mathbb R^n$. If the indicator function $\mathbb 1_S$ is integrable. We say that $S$ is rectifiable and we define the ($n$-dimensional) volume of $S$ by the equation $$V(S) = \int_{\mathbb R^n} {\mathbb 1_S}.$$

Furthermore a subset $S$ of $\mathbb R^n$ is rectifiable if and only if $S$ is bounded and $\partial S$ has measure zero. So basically its a set whose Jordan content is defined i.e it is a Jordan-measurable set.