Real Algebraic and Analytic Geometry
Submission: 2003, October 31.
Given an o-minimal structure M which expands a field, we define, for each positive integer d, a real valued additive measure on a Boolean algebra of subsets of M^d and we prove that all the definable sets included in the finite part Fin(M^d) of M^d are measurable. When the domain of M is IR we obtain the Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets. Our measure has good logical properties, being invariant under elementary extensions and under expansions of the language.
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