I have a few amateur, naive questions about the Banach-Tarski decomposition. I have a few amateur, naive questions about the Banach-Tarski decomposition. Let’s take a unit sphere (B1), and perform the decomposition so that we have a new sphere with twice the volume (B2). I realize that this transformation involves the use of the axiom of choice, and that B2 is composed of non-measurable sets.

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*If B2 is not measurable, what is the meaning of the “twice the volume” statement?


*Let’s assume the B1 is the standard closed unit ball. It is certain characteristics. It is measurable, it is a metric space, and locally homeomorphic to R^3. Topologically, B1 is compact, locally compact, and paracompact. It is connected, simply connected, locally connected, path- and arc- connected. Which of these characteristics change when considering B2?
I imagine there is a bijective mapping from B1 to B2. I also suppose B2 is a Vitali set.
Thank you!
 A: Nothing changes between $B_1$ and $B_2$ really. The statement of Banach-Tarski is (in the formulation I learnt from Stan Wagon's book The Banach-Tarski paradox): let $A,B$ be two bounded subsets of $\Bbb R^3$ both with non-empty interior. Then $A$ and $B$ are equi-decomposable ($A = \bigcup_{I=1}^n A_i$ and $B = \bigcup_{i=1}^n B_i$ where $n \in \Bbb N$ and for each $i$ we have a linear map $g_i \in SO_3$ (a rotation even) so that $g_i[A_i]=B_i$).
So your question is about when we take two balls of different volumes, but the volumes have nothing to do with a measure per se; the old Greeks already knew the formula for computing the volume of a sphere from its radius, $\frac{4}{3}\pi r^3$. But the above theorem allows us to have one sphere $B_1$, and cut it up in finitely many pieces using AC (so very non-constructively) and transform (rotate etc) those pieces and take the union of the transformed ones and get another sphere with larger radius(or smaller radius, if you like; or build two disjoint spheres or a cube etc.)
It's the pieces that are strange, not the end results. Your $B_1$ and $B_2$ are just homeomorphic and topologically indistinguishable. Banach showed in essence that the group of 3d-rotations is "weird" (or as Wagon calls it "paradoxical"). The relation with measures is that is shows that under AC there can be no finitely additive measure on $\Bbb R^3$ that measures all subsets and is rotation invariant while having the volume of a sphere have the known value. So it's not a paradox in that sense, it's just counterintuitive. Read Wagon's book to really understand Banach-Tarski, it's very good IMO.
