Finally we treat the case that Y is VF. By Theorem 5.3.1 and by the already treated cases that Y = RF n and Y = VG, it is enough to treat the case that W K,x is a single open ball in K for each K and each x ∈ X K . Moreover, on these balls, we may assume that the functions like h as in (3.2.1) that appear in the build-up of f have the Jacobian property, and that all other build-up **data** of f factors through the projection W → X. But then calculating the integral is easy, and goes as follows. By the Jacobian property for h, the set h K (x, W K,x ) is an open ball, say, of valuative radius n K ∈ Z. If n K ≤ 0, then the integral of ψ(h K (x, y)) over y ∈ W K,x equals zero for each ψ in D K . If n K > 0, then ψ(h K (x, y)) is constant on W K,x , say, equal to ξ K,ψ,x , and hence the integral of ψ(h K (x, y)) over y ∈ W K,x equals the volume of the ball W K,x , say q m K K , times ξ K,ψ,x . By Lemma 4.6.2 and by the already treated cases that Y = RF n , we may suppose that h factors through the projection W → X, so that q m K K ξ K,ψ,x lies in C exp (X) by definition of C exp -functions and we are done. (1) For every definable set X, C(X) contains the characteristic function of every definable set A ⊂ X. (2) The collection is stable under integration, namely, for any X and Y and any f ∈ C(X × Y ) such that for each K and each x ∈ X K , the function y → f K (x, y) is integrable over Y K , the function
(K, x) → y∈Y K f K (x, y)|dy|
lies in C(X) (with our usual product measure on Y K ).