$\DeclareMathOperator{\graph}{\operatorname{graph}}$
I would like to know if, given $f\in W^{1,2}(\mathbb{R}^n,\mathbb{R})$, it is true that we can always cover $\graph(f)\subset\mathbb{R}^{n+1}$ with the images of countably many Lipschitz maps $g\_k:\mathbb{R}^n\to\mathbb{R}^{n+1}$ and a set $E$ with $\mathscr{H}^n(E)=0$, i.e.
$$\graph(f)\subset \bigcup\_{k\in\mathbb{N}} g\_k(\mathbb{R}^n) \cup E$$
(which is the definition of $\graph(f)$ being countably rectifiable).
Any hint, reference or additional conditions for this to be true are very much appreciated.



Is the support of a Sobolev function a varifold?
Yes if you choose a suitable representative of a Sobolev function.


**Lemma.** *Let $f\in W^{1,p}(\mathbb{R}^n)$, $1\leq p<\infty$. Then for every $\epsilon>0$, there is a Lipschitz function $g:\mathbb{R}^n\to\mathbb{R}$ such that
$$
|\{ x\in\mathbb{R}^n:\, f(x)\neq g(x)\}|<\epsilon.
$$*
The proof follows from the pointwise inequality
$$
|f(x)-f(y)|\leq C|x-y|(M|\nabla f|(x)+M|\nabla f|(y)),
$$
where $M$ stands for the Hardy-Littlewood maximal function.
$\Box$


The lemma implies that you can find countably many Lipschitz functions $g\_i:\mathbb{R}^n\to\mathbb{R}$ such that such that the set
$$
E:=\mathbb{R}^n\setminus \bigcup\_{i=1}^\infty\{x\in\mathbb{R}^n:g\_i(x)\neq f(x)\}.
$$
has measure zero.
If you redefine $f$ on $E$ so that the function $f$ is constant on that set (that alters $f$ on a set of measure zero which is okay, because Sobolev functions are defined a.e.) it follows that graps of the functions $g\_i$ and a graph of a constant function cover the graph of $f$.


For more details see for example Lemma on p. 96 in:


**P. Hajłasz,** Change of variables formula under minimal assumptions. *Colloq. Math.* 64 (1993), no. 1, 93–101.


