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. I The extensions given by Hahn-Banach don't know if such a measure is uniqueseem to have any uniqueness properties, but so I doubt itthis measure is unique.
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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. I don't know if such a measure is unique, but I doubt it. |
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