Let $\rho(\cdot, \cdot)$ be any distance metric on the set $\mathbb{R}^{N+1}$ and let $S^N = \{\mathbf{x} \in \mathbb{R}^{N+1}: \rho(\mathbf{0},\mathbf{x}) = 1\}$ be the unit N-sphere.
For a fixed $k \in [0, 2]$, let $M(\bar{\mathbf{x}}) = \{\mathbf{x} \in S^N : \rho(\bar{\mathbf{x}}, \mathbf{x}) \leq k\} $ denote the set of points on the sphere $S^N$ that are within a distance of $k$ from the point $\bar{\mathbf{x}}$.
How do I rigorously show that $\lambda(M(\bar{\mathbf{x}}))$ = $\lambda(M(\bar{\mathbf{y}}))~\forall \bar{\mathbf{x}}, \bar{\mathbf{y}} \in S^N$? Here, $\lambda(\cdot)$ denotes the Lebesgue measure of its argument.
Is there a way I can define rotational invariance for the Lebesgue measure on an $N-$sphere for a given distance metric?