Borel Ruziewicz problem

The original Ruziewicz problem was posed by Ruziewicz in the early 1900s and asked whether Lebesgue measure is the unique finitely additive, isometry-invariant measure on the Lebesgue measurable subsets of $\mathbb{S}^n$ (for $n \geq 2$) which assigns measure $1$ to $\mathbb{S}^n$. It was answered in the affirmative by results of Sullivan, Margulis and Drinfeld. The Borel Ruziewicz problem modifies the original Ruziewicz problem by only requiring the measure to be defined on the Borel measurable sets rather than all Lebesgue measurable sets.

It is known that the Ruziewicz problem (and hence also the Borel Ruziewicz problem) has a negative answer for $\mathbb{S}^1$, hence the restriction to $n \geq 2$ in the problem statement.