[FOM] Banach Tarski Paradox/Line

W.Taylor at math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Tue Nov 29 06:05:46 EST 2011

Quoting "K. P. Hart" <K.P.Hart at tudelft.nl>:

> Banach proved that Lebesgue measure on the real line has
> a finitely additive extension to the family of all subsets.
> This extension is also invariant under isometries.

It is significant that it is not necessarily countably additive,
i.e. a "true measure" (in today's terms).

This recalls to my mind a strange oddity, a collection of results
proved by Solovay in the early 70's.

It turns out, that if one wishes to extend the simple concept
of "length" from plain intervals to as large a collection of
subsets of R (equivalently [0,1] ) as possible, retaining
at least finite additivity, one may get ANY THREE of
the following properties, but NOT all four:-

1) countable additivity;
2) all subsets are measurable;
3) it is translation invariant;
4) axiom of choice holds.

I always thought this was cool in that it is the only result
of its type - the 3-out-of-four-ness of it; unless anyone knows
of any similar result with (at least) a 2-out-of-3-ness feature.

Does anyone know of any such case?

It is also strange in that it gives AC the appearance of a purely
measure-theoretic property; though this is perhaps less surprising,
as many algebraic and topological equivalents are by now also known.

-- wfct

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