FOM: Measures of semi algebraic sets
Harvey Friedman
friedman at math.ohio-state.edu
Sun Dec 14 12:07:25 EST 1997
This is prompted by Steve Simpson's posting of 14 Dec 1997 13:07, which in
turn is a reply to Feferman, 8:54AM 12/14/97.
There is a much more draconian restriction for which the kind of pathology
you speak of is not present. This is the semi algebraic subsets of
Euclidean space. These are the Boolean combinations of graphs of polynomial
inequalities with real coeffiencts and real variables. They can be
normalized as the finite unions of solutions to systems of polynomial
inequalities (< and =). By Tarski, they are also the first order definable
relations of several variables over the field of real numbers. In any
dimension, the measure of a set is the same as the measure of its
topological closure. This means that for bounded sets, one can use Jordan
content.
It would seem that measure is pretty well behaved for semi algebraic sets.
QUESTION: In what sense is measure "entirely well behaved" for semi
algebraic sets?
Also, a specific question comes to mind:
QUESTION: Can you decide whether or not two semi algebraic sets with
rational coefficients have the same measure? Can you decide whether or not
a semi algebraic set with rational coefficients has finite measure?
Perhaps one can show that one of these questions is equivalent to some more
basic question which may involve too much unsolved number theory. One might
be able to formulate a basic decision problem, and study related problems
in terms of algorithmic reducibility to it.
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