TITLE: Banach-Tarski Paradox for the unit interval?
QUESTION [13 upvotes]: Does there exist such a map $\phi: [0, 1] \rightarrow S^2$ which determines some sort of a Banach-Tarski decomposition for the unit interval $[0, 1]$? I did read through Stan Wagon's The Banach-Tarski Paradox but the language is somewhat terse. I did read somewhere that Felix Hausdorff proved that one can chop up the unit interval into countably many pieces, slide the pieces around, and fit them together into the interval $[0, 2]$. Not sure of any journals that describe the exact proof. Is it possible to find $\phi$ explicitly perhaps in terms of the middle-third Cantor set?

REPLY [4 votes]: To get such a paradox using finitely many pieces, as in the Banach-Tarski case, one must move beyond isometries. But if one uses measure-preserving transformations, such a paradox does exist on the line. Reviewing my book on the BTP I find Theorem 7.9 which shows that there is a paradoxical decomposition of the interval using transformations preserve Lebesgue measure. The main point is that there is a Lebesgue-measure preserving function from the interval to the sphere, and that can be used to move the Banach-Tarski Paradox down to the line.
And there is von Neumann's paradox (Thm 7.12) which provides a finitary paradoxical decomposition of the line using epsilon contractions, for any positive epsilon. Since such a contraction is a linear fractional transformation it shrinks Lebesgue measure, so this can be considered paradoxical. Both of these use the Axiom of Choice.
Stan Wagon