Let $d\in\mathbb N$ and $M$ be a $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$ with boundary.
In the context of the trace operator, I've seen the usage of the $L^p$-space $L^1(\partial M)$. What is this space?
Usually, if $\Omega\in\mathcal B(\mathbb R^d)$, $L^1(\Omega)$ is a shorthand notation for the space $L^1\left(\left.\lambda^{\otimes d}\right|_\Omega\right)$, where $\lambda$ denotes the Lebesgue measure on $\mathcal B(\mathbb R)$.
But I doubt that $L^1(\partial M)$ is really denoting $L^1\left(\left.\lambda^{\otimes d}\right|_{\partial M}\right)$ here, since $\lambda^{\otimes d}(\partial M)=0$ and hence this space is rather boring.
My most probable guess is that $L^1(\partial M)$ has to be understood as a shorthand notation for $L^1(\sigma_{\partial M})$, where $\sigma_{\partial M}$ denotes the surface measure on $\mathcal B(\partial M)$.
Am I right? If not, what's denoted by $L^1(\partial M)$?