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|qx , where |μ|qx 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). © 2013 Elsevier Inc. All rights reserved