Why is there a need to define $L^2 (\partial \Omega)$ with $\Omega$ being a Lipschitz, open and connected set, when we already know how to define $L^2(\Omega)$?
$L^2(\Omega)=\{u: \int_{\Omega} u^2<\infty \}$ with $u:\Omega\rightarrow \mathbb{R}$.
Couldn't we just do $L^2(\partial \Omega)=\{u: \int_{\partial \Omega} u^2<\infty \}$ with $u:\partial \Omega\rightarrow \mathbb{R}$.
I ask this question, since I'm reading a book on PDEs and, before talking about Traces of generalized functions, the author takes a moment to define $L^2(\partial \Omega)$.
Edit to reflect some comments.
If we use a different measure $\mu$ for $\partial \Omega$, then wouldn't the following still be valid?
$$L^2(\partial \Omega)=\{u: \int_{\partial \Omega} u^2(x) d\mu<\infty \}$$ with $u:\partial
\Omega\rightarrow \mathbb{R}$