From: Per Erik Manne <Per.Manne@nhh.no>
Newsgroups: sci.math
Subject: Re: finitely additive measures on R, R^2
Date: Mon, 02 Nov 1998 13:02:37 +0100

theodore hwa wrote:

> The Banach-Tarski paradox in dimensions >=3 implies that for n>=3 it's
> impossible to define a finitely additive measure on all subsets of R^n
> that's invariant under translations and rotations.  But for n=1,2 no such
> paradox exists and I understand that it's possible, perhaps using AC, to
> define such a measure in dimensions 1 and 2.  It will not be countably
> additive.  How does one construct such a measure?  Are there any good
> references on this sort of thing?
>

Stan Wagon: The Banach-Tarski Paradox (Cambridge University Press 1985).

Part II is about finitely additive measures, or the nonexistance of
paradoxial
decompositions. I never read this, being more interested in Part I. A quick
glance seems to indicate that given a group G which acts on a set X, one
tries to construct a finitely additive measure on G, defined on all subsets,
and then to transfer it to X. If such a finitely additive measure (with some
extra conditions) exists on G, then G is called amenable. In theorem 10.4
one proves with the aid of AC that all abelian groups are amenable, and
that the class of amenable groups is closed under certain operations
(subgroups, quotients, etc.). It follows that all solvable groups are
amenable,
and since the isometry groups of R^1 and R^2 are solvable, we are done
(almost).
--
Bergen,
Per Manne
==============================================================================

From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: finitely additive measures on R, R^2
Date: 2 Nov 1998 21:41:05 GMT

In article <71bb0i$ast$1@nntp.Stanford.EDU>, hwatheod@leland.Stanford.EDU (theodore hwa) writes:
|> The Banach-Tarski paradox in dimensions >=3 implies that for n>=3 it's
|> impossible to define a finitely additive measure on all subsets of R^n
|> that's invariant under translations and rotations.  But for n=1,2 no such
|> paradox exists and I understand that it's possible, perhaps using AC, to
|> define such a measure in dimensions 1 and 2.  It will not be countably 
|> additive.  How does one construct such a measure?  Are there any good
|> references on this sort of thing?

Try "Invariant Means on Topological Groups" by Frederick P. Greenleaf,
Van Nostrand 1969.

Here's a slightly more "explicit" construction than Per Erik Manne's, in
the case n=1.  We want a linear functional phi on the space B(R) of bounded
real-valued functions on R such that 
1) phi(1) = 1 
2) if f >= 0 then phi(f) >= 0
3) phi is invariant under translations T_x(f)(t) = f(t-x)
4) phi is invariant under reflection r(f)(t)=f(-t)

Lemma:  For any f_1, ..., f_n in B(R) and x_1, ..., x_n in R, 
let g = sum_{i=1}^n (f_i - T_{x_i} f_i).  Then inf_{x in R} g(x) <= 0.
Proof: Consider S = the sum of  g(sum_i j_i x_i) where
each j_i ranges over the integers 1 to p.  Express it in terms of the f_i,
and all terms cancel except for those involving f_i(sum_k j_k x_k) where 
j_i = 0 or p.  Thus |S| <= K p^(n-1) for some K.  Since there are
p^n j's, there is some x = sum_i j_i x_i with g(x) <= K/p.  Take p ->
infinity to get the result.

Now let M be the closed span of the functions f - T_x f (f in B(R), x in R)
and U = { g in B(R): inf_{x in R} g(x) > 0 }.  By one of the forms of the
Hahn-Banach Theorem there is a linear functional phi on B(R) such that
phi = 0 on the linear subspace M and phi > 0 on the convex set U, and we
can normalize phi so phi(1) = 1.  It is then easy to see that (1) to (3)
are satisfied.  To get (4) in addition, we replace phi by 
psi(f) = (phi(f) + phi(r(f)))/2.

To define the measure mu, take mu(A) = phi(I_A) where I_A is the indicator
function of A (I_A(x) = 1 if x in A, 0 otherwise).

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2