Let $\Sigma$ be a submanifolds $C^{1}$ of $\mathbb{R}^{n}$ $k$-th dimensional and let $g : \Omega \subseteq \mathbb{R}^{k} \longmapsto \Sigma \subseteq \mathbb{R}^{n}$ $C^{1}$ with $g$ injective and $\text{D}g(x)$ injective.
Given $A \subseteq \Omega$ Borel set or Lebesgue measurable we define $H_{k}(g(A)) := \int_{A} \sqrt{\text{det}(\text{D}g(x)^{T}\text{D}g(x))} dH_{k}$.
$H_{k}$ defines a well defined measure invariant by $C^{1}$ reparametrization.
Now, let $E \subseteq \partial B(0,1,\mathbb{R}^{n+1})$ be a Borelian set.
We set $m(E) := \frac{n+1}{r} \lambda_{n+1}([0,1]E)$, where $\lambda_{n+1}$ is the Lebesgue $n+1$ dimensional measure and $[0,1]E = \left\lbrace tx : t \in [0,1] , x \in E\right\rbrace$.
I'd like to prove that the measure defined are equivalent.
I know there is an important characterization of uniformely distributed measure by Christensen, cited here, but since I don't have the knowledge required to understand it.
I determined a parametrization of $\mathbb{S}^{n}$, with polar coordinates I tried to compute $\text{det}(\text{D}g(x)^{T}\text{D}g(x))$, without success.
The idea was to compute the determinant cited above and compare with the integral given by the defition of $m(E)$, but I was unable to do so. Was my intetion correct ?
Any help or hint would be appreciated.
