TITLE: Sobolev Space $W_0^{1,p}(I)$ and the boundary of $I$
QUESTION [1 upvotes]: Given $I \subset\mathbb{R}$ an open interval, the Sobolev Space $W_0^{1,p}(I)$ is defined as $W_0^{1,p}(I)=\overline{C^1_c(I)}^{W^{1,p}(I)}$ (The closure of $C^1_c(I)$ on the space $W^{1,p}(I)$) .
There is a basic result that tells us:

Given $u \in W^{1,p}(I)$, then $u \in W_0^{1,p}(I)$ if and only if $u=0$ on $\partial(I)$.

Considering the fact that the boundary of $I$ is at most consisting of two points and that two points is a set of meassure zero, Is the boundary the usual topological boundary? Does this result mean there is some representative $\tilde{u}$ of the equivalence class of $u$ which actually banishes at $\partial(I)$?

REPLY [0 votes]: The points do not have measure zero. On the (topological) boundary you typically have to consider the (so called Hausdorff-) measure of one dimension less (for, e.g., defining equivalence classes). And the zero dimensional measure is the "counting measure". Every point has measure one.
{"set_name": "stack_exchange", "score": 1, "question_id": 1742625}