# reference request – Hausdorff Dimension e Surface Measure

Could someone please indicate me some reference that contains the proof of the following theorem?

Below $$mathcal{H}^n$$ denotes the $$n$$-dimensional Hausdorff outer measure in $$mathbb{R}^n$$.

Theorem: Let $$Msubset mathbb{R}^N$$ be a $$k$$-dimensional manifold of class $$C^1$$, $$1leq kleq N$$.

1. Let $$varphi$$ be a local chart, that is, $$varphi:Ato M$$ is a function of class $$C^1$$ for some open set $$A subset mathbb{R}^k$$ such that $$nabla varphi$$ has maximum rank $$k$$ in $$A$$. Define $$g_{ij}:=frac{partial varphi }{partial y_i}!cdot !frac{partial varphi }{partial y_j}$$, where $$cdot$$ is the inner product in $$mathbb{R}^N$$. Then $$varphi (A)$$ has Hausdorff dimension $$k$$ and

$$mathcal{H}^k(varphi(A))=int_A sqrt{det g_{ij}(y)}dy$$

1. $$M$$ has Hausdorff dimension $$k$$ and that $$mathcal{H}^k(M)$$ is the standard surface measure of $$M$$.

I found this theorem in the file “Measure and Integration” (pg 9).

I searched for some reference that contains the proof of the above theorem but couldn’t find it.

I am posting this request here since a similar question on Mathematics Stack Exchange didn’t receive an answer!