I am studying PDEs and in some books (Folland, **Introduction to Partial Differential Equations** and Evans, **Partial Differential Equations**), I found an integral integrated by the **surface measure** on a $C^k$ hypersurface of $\mathbb{R}^n$.

1) Is there any reference to see how this measure is defined?

2) What is the weakest assumption for a subset $M\subset \mathbb{R}^n$ in order that the surface measure is defined on $M$, for example: $C^k$ for which values of $k$?

3) When M is compact, is the surface measure different from the surface integral defined in Spivak's **Calculus on Manifolds**?

4) When one considers $M$ as a Riemannian manifold with the induced metric,
is this surface measure equal to the Riemannian measure (as defined, for example, in GriGor'yan, **Heat kernels and Analysis on Manifolds**)?

"some book"$\endgroup$ – Praphulla Koushik Feb 15 at 8:47