Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characteristic as a finitely-additive measure. In the presence of a metric, it can be extended to continuous integrands via the Gauss–Bonnet theorem. It was introduced independently by Pierre Schapira and Oleg Viro in 1988, and is useful for enumeration problems in computational geometry and sensor networks.
Euler integration for constructible functions
Euler calculus begins from the observation that the Euler characteristic obeys one of the main properties of a measure: . As a result, for a suitably restricted class of functions, it is possible to define an integral with respect to this measure. One begins by selecting an o-minimal structure of definable sets in the topology, for instance, semialgebraic or subanalytic sets. The class of constructible functions consists of those functions such that is definable for all . A definition of the Euler integral follows:
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