The aim of this note is twofold. The first is to obtain a characterization of relative constructible sheaves under a natural condition on the parameter space. The second is to introduce and study the notions of relative constructible functions and relative Euler-Poincar\'e index. We prove that the relative Euler-Poincar\'e index provides an isomorphism between the Grothendieck group of the derived category of complexes with bounded relative $\mathbb R$-constructible cohomology and the ring of relative constructible functions.PDF: A note on the ring of relative constructible functions.pdf
