On the Vitali set

Question

It is well known that the Vitali set is unmeasurable. When given a measure space $(\mathbb{R},\mathcal{M},\mu)$, can we say that the Vitali set as a subset of $\mathbb{R}$ is never measurable regardless of the choice of the measure $\mu$?

I know the Vitali set is not Lebesgue measurable, but is the Vitali set unmeasurable in every measure space?

Answer 1

The proof that a Vitali set is not Lebesgue measurable relies on the fact that Lebesgue measure is invariant under translation. Unless the measure space $(\mathbb R, \mathcal M, \mu)$ has the property:

if $A \in \mathcal M$ and $\mu(A) > 0$, then $A+x \in \mathcal M$ and $\mu(A+x) > 0$ for all $x$,

then we cannot expect that argument to work.