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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)?

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    $\begingroup$ 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. $\endgroup$
    – juan
    Feb 15, 2019 at 8:47
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    $\begingroup$ May be you should specify what book it is rather than saying "some book" $\endgroup$
    – Praphulla Koushik
    Feb 15, 2019 at 8:47
  • $\begingroup$ @PraphullaKoushik On Evans and Follands Books on PDEs they say surface measure. Thanks! $\endgroup$
    – Studentmathever
    Feb 15, 2019 at 16:32
  • $\begingroup$ Please consider adding in the question itself.. $\endgroup$
    – Praphulla Koushik
    Feb 15, 2019 at 17:27

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