[FOM] Banach Tarski Paradox/Line

[David Diamondstone]

Wikipedia says that the Banach-Tarski paradox is impossible in two dimensions, because there is a finitely-additive measure on the plane, invariant under isometries, such that the measure of the unit square/disk is positive and finite. Since the Banach-Tarski paradox implies the existence of non-measurable sets, the existence of a finitely-additive measure with no non-measurable sets makes it impossible. Unfortunately Wikipedia declines to provide either details or a reference, but maybe it is easy to show such a measure exists? I would be interested in seeing an example/construction of such a measure.

Assuming Wikipedia is correct, it would also be impossible in 1 dimension, for the same reason.


On Nov 28, 2011, at 10:27 AM, pax0 at seznam.cz wrote:

> Is the Banach Tarski paradox provable for the unit real interval; 
> i.e. is there a possibility for duplicating [0,1].
> If not, where is the obstacle?
> Jan Pax
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