In measure theory, the Euler measure of a polyhedral set equals the Euler integral of its indicator function.

The magnitude of an Euler measure

By induction, it is easy to show that independent of dimension, the Euler measure of a closed bounded convex polyhedron always equals 1, while the Euler measure of a d-D relative-open bounded convex polyhedron is ${\displaystyle (-1)^{d))$.^[1]