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The $n$-dimensional affine group over the integers is the group $\mathcalG_n$ of all affinities on $\mathbb R^{n}$ which leave the lattice $ \mathbbZ^{n}$ invariant. $\mathcal G_n$ yields a geometry in the classical sense ofthe Erlangen Program. In this paper we construct a $\mathcal G_n$-invariantmeasure on rational polyhedra in $\mathbb R^n$, i.e., finite unions ofsimplexes with rational vertices in $\mathbb R^n$, and prove its uniqueness.Our main tool is given by the Morelli-W{\l}odarczyk factorization of birationaltoric maps in blow-ups and blow-downs (solution of the weak Oda conjecture).

Measure theory in the geometry of $GL(n,\mathbb Z) \ltimes \mathbb Z^{n}$