Euler integration over definable functions

We extend the theory of Euler integration from the class of constructible functions to that of "tame" real-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it is not a linear operator); however, it has a compelling Morse-theoretic interpretation. In addition, we show that it is an appropriate setting in which to do numerical analysis of Euler integrals, with applications to incomplete and uncertain data in sensor networks.