NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/research/663

Text File | 1993-01-27 | 1.8 KB | 48 lines

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From: [e-mail] (Cameron Laird)
Subject: Set-free measure theory
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Date: Tue, 26 Jan 1993 14:47:10 GMT
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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
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