One can define $L^p$ spaces for measure spaces with a given measure. Is a non-compact (i.e., it has a boundary) bounded Riemannian manifold a measure space? I am thinking of the manifold $(0,T) \times S$ where $S$ is a $n-1$ dimensional hypersurface embedded in $\mathbb{R}^n$.
I want to know so I can use the $L^p$ theory already available. Are there any technicalities I need to worry about? Basically I want density of continuous functions on a manifold that I have in mind. Thanks.
As studiosus explained, the canonical measure is available, given by the volume (pseudo-)form.
To show that continuous (or smooth) functions are dense in $L^p(M)$, use a partition of unity into compactly supported functions, each of which is supported in a coordinate patch (an open set diffeomorphic to an open set in $\mathbb R^n$). This makes it possible to transfer the density result from $\mathbb R^n$ to the manifold.