Finitely additive translation invariant measure on $\mathcal P(\mathbb R)$

Banach-Tarski poses a problem for existence of measures that are invariant under all rigid motions, not just translation. The existence of finitely additive translation-invariant measures that agree with Lebesgue measure on Lebesgue-measurable sets is a consequence of the Hahn-Banach theorem. This is exercise 21 in chapter 10 of Royden's Real Analysis. The extensions given by Hahn-Banach don't seem to have any uniqueness properties, so I doubt this measure is unique.

In n=1 it exists, but far from unique. This is in Hewitt and Stromberg. I believe they show there are 2^c different extension.