Euler Integral

Euler integration was defined by Schanuel and subsequently explored by Rota, Chen, and Klain. The Euler integral of a Function $f:\Bbb{R}\to\Bbb{R}$ (assumed to be piecewise-constant with finitely many discontinuities) is the sum of

$$f(x) - {\textstyle{1\over 2}}[f(x_+)+f(x_-)]$$

over the finitely many discontinuities of $f$. The $n$-D Euler integral can be defined for classes of functions $\Bbb{R}^n\to\Bbb{R}$. Euler integration is additive, so the Euler integral of $f+g$ equals the sum of the Euler integrals of $f$ and $g$.

See also Euler Measure