It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{1,p}(\Omega)=W^{1,p}(\Omega\backslash\Sigma)$ for $p\in (1,+\infty)$.
Question: Assume $F$ and $\Omega$ satisfy the conditions prescribed above. In this case, shall we have the following equality $W^{1,p}_{0}(\Omega)=W^{1,p}_{0}(\Omega\backslash\Sigma)$ ?
Thanks for any suggestion.