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I've been reading about paradoxical sets, mainly paradoxical subsets of the plane. As a consequence of this, I've been reading a couple of G.A. Sherman's papers on the subject. In his paper "Properties of Paradoxical Sets in the Plane," an interesting result is that any subset of the plane with nonempty interior is not paradoxical (this is very interesting in that it is contrary to the the 3D analog where all bounded subsets with nonempty interior have to be paradoxical by the Banach-Tarski Paradox). However, to prove this, he used a total, finitely-additive, isometry-invariant extension of Lebesgue Measure, which he calls a Banach measure. His only references for this seem to be a paper by Banach, which is in French, and a theorem from "The Banach-Tarski Paradox" by Stan Wagon, which doesn't seem to mention Banach measures by name, nor does any of the surrounding material expound on this.

I think I've been able to understand and fill in the details for most of his proofs for the main theorems, but I want to solidify my understanding by getting to know Banach measures better. Are there any textbooks or references that make specific use of Banach measures? What theorems can we take from the Lebesgue measure and put in terms of Banach measures? I assume, since he uses it, that nonempty interior implies positive measure for Banach measures, like it does for the Lebesgue measure, but is there anything else?

Brian
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    Seems likely that you've already googled "Banach measures" for yourself, but this google.books link looks as though it might be helpful: http://books.google.co.uk/books?id=rbCmt-2NxtIC&pg=PA120&lpg=PA120&dq=%22banach+measures%22&source=bl&ots=_hRnRyHcS4&sig=6c6XlNx0FoE_DJA63ayA-xRfFzE&hl=en&sa=X&ei=c26xUqWKO4Ob0AWkpIDIAg&ved=0CFEQ6AEwBQ#v=onepage&q=%22banach%20measures%22&f=false – postmortes Dec 18 '13 at 09:49
  • Thank you, I'll check it out. – Brian Dec 18 '13 at 18:09

1 Answers1

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Chapter 10 of my book, The Banach-Tarski Paradox, discusses the existence of and construction of such measures in full detail: Theorem 10.8. When the group in question is Abelian (Banach's original work) or solvable (von Neumann) or, more generally, amenable, then a G-invariant measure on all sets exists. That is why the BTP does not exist in the line and plane.

I and G. Tomkowicz are working on revising my old (1985) book and we too have been reading the Sherman papers with interest and hope to include this result.

Stan Wagon

stan wagon
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  • Thank you very much for the reply. I've had your book on the to-read list for awhile now, but now I even have more than just curiosity to motivate me. I think it was the technical language used that threw me off, as I was expecting to see something with "Banach measure" explicitly written rather than stuff about an extension of Lebesgue measure that is invariant under an amenable group of isometries. Of course, now that I've had another look at it, I see that this is exactly what a Banach measure would be. Thank you, again. – Brian Dec 18 '13 at 20:00