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It is well known that the Lebesgue measure is the unique (up to a multiplicative constant) sigma-finite Borel measure on $\mathbb{R}^d$ which is translation invariant. I am wondering if a similar ...

Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively. Question ...