First order tameness of measures

We develop a general framework for measure theory and integration theory that is compatible with o-minimality. Therefore the following natural definitions are introduced. Given are an o-minimal structure M and a Borel measure μ on some Rn. We say that μ is M-compatible if there is an o-minimal expansion of M such that for every parameterized family of subsets of Rn that is definable in M the corresponding family of μ-measures is definable in this o-minimal expansion. We say that μ is M-tame if there is an o-minimal expansion of M such that for every parameterized family of functions on Rn that is definable in M the corresponding family of integrals with respect to μ is definable in this o-minimal expansion. We substantiate these definitions with existing and many new examples. We investigate the Lebesgue measure in their light. We prove definable versions of fundamental results such as the theorem of Radon–Nikodym, the Lebesgue decomposition theorem and the Riesz representation theorem. They allow us to describe explicitly compatible and tame measures and to classify them in dimension one.