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In article <C1GuEp.M99@sugar.neosoft.com> claird@NeoSoft.com (Cameron Laird) writes:

   Conjecture:  the literature already includes a proof
   that there is nothing novel about "set-free measures",
   that is, they all have realizations as (extended)
   measures.  Does anyone know?

Apparently, you don't require the boolean algebra to be a sigma algebra.
In that case, the Stone representation theorem seems to answer your
question: every boolean algebra is isomorphic to the algebra of
clopen (i.e. simultaneously open and closed) subsets of a totally
disconnected compact Hausdorff space.

If you want to work with sigma algebras and want the representation
to be a sigma homomorphism, the Stone representation doesn't work anymore.
I'm sure a lot of work has been done on the representation of sigma
algebras as sigma algebras of subsets, but I don't know the results
on it.

If your Boolean algebra is atomic, you might get a suitable representation
of it as subsets of the set of its atoms.

If you look at Borel sets modulo sets of measure zero, you don't have
sets any more. This is not an atomic boolean algebra in general.


Allan Adler
ara@altdorf.ai.mit.edu