Set-free measure theory

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3 total messages Started by Cameron Laird Tue, 26 Jan 1993 14:47

#192

Author: Cameron Laird
Date: Tue, 26 Jan 1993 14:47

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I propose this definition:  a "set-free measure" is
a triple <X, m, T> with
1.  X a lattice (boolean algebra),
2.  m a function defined on the members of X, and
3.  T an equivalence relation on X, for which
4.  m(x + y) = m(x) + m(y) - m(x.y) and
5.  xTy implies m(x) = m(y).

The natural model for such an object is conventional
measure theory, with X a collection of Borel sets, or
measurable sets, taken from an abelian group, and T
translation (that is, read xTy as "x is a translate of
y").  Are there any (non-trivial) others?  I realize I
haven't constrained the range of m (finite, includes
zero, non-negative, ...), or adjoined a countable
version of 4; if doing so makes the problem more
interesting, then I'd want to know.

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?

I write "set-free" to emphasize the historical attention
directed away from the point-set flavor typical of measure
theory, and toward a categorical attention to the action
of the function m.
-- 

Cameron Laird
claird@Neosoft.com (claird%Neosoft.com@uunet.uu.net)	+1 713 267 7966
claird@litwin.com (claird%litwin.com@uunet.uu.net)  	+1 713 996 8546

Re: Set-free measure theory

#196

Author: Gerald Edgar
Date: Tue, 26 Jan 1993 21:11

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In article <C1GuEp.M99@sugar.neosoft.com> claird@NeoSoft.com (Cameron Laird) writes:
>I propose this definition:  a "set-free measure" is
>a triple <X, m, T> with
>1.  X a lattice (boolean algebra),
>2.  m a function defined on the members of X, and
>3.  T an equivalence relation on X, for which
>4.  m(x + y) = m(x) + m(y) - m(x.y) and
>5.  xTy implies m(x) = m(y).
>

A place to start your reading is Chapter X of Birkhoff's _Lattice Theory_
(3rd edition).  You have described a lattice with valuation (if you
omit 3 and 5).

-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)

Re: Set-free measure theory

#197

Author: Allan Adler
Date: Wed, 27 Jan 1993 04:34

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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

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