# Definition of the surface measure in some books

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)?

• The more general definition is by Hausdorff measure. You only need a metric space to define it. A good reference will be the book by Folland, Real Analysis, where the connection with the more elementary definitions is done. – juan Feb 15 at 8:47
• May be you should specify what book it is rather than saying "some book" – Praphulla Koushik Feb 15 at 8:47
• @PraphullaKoushik On Evans and Follands Books on PDEs they say surface measure. Thanks! – Studentmathever Feb 15 at 16:32
• Please consider adding in the question itself.. – Praphulla Koushik Feb 15 at 17:27