A notion of integration using the Euler characteristic as a measure.
Yuliy Baryshnikov, Robert Ghrist, Euler integration over definable functions (pdf)
С. М. Гусейн-Заде, Интегрирование по отношению к эйлеровой характеристике и его приложения, УМН, 65:3 (393) (2010) 5–42, link; Eng. transl. Gusein-Zade, Integration with respect to the Euler characteristic and its applications, Russ. Math. Surv. 65:3,2010, 399-432, link, MR2011k:14011
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