Lebesgue classes and preparation of real constructible functions

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any q>0 and constructible functions f and μ on E×Rn, we prove a theorem describing the structure of the set {(x,p)∈E×(0,∞]:f(x,⋅)∈Lp(|μ|xq)}, where |μ|xq is the positive measure on Rn whose Radon–Nikodym derivative with respect to the Lebesgue measure is |μ(x,⋅)|q:y↦|μ(x,y)|q. We also prove a closely related preparation theorem for f and μ. These results relate analysis (the study of Lp-spaces) with geometry (the study of zero loci).