### A Bochner type theorem for inductive limits of Gelfand pairs

In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs $(G,K)$ defined as inductive limits of increasing sequences of Gelfand pairs ${(G\left(n\right),K\left(n\right))}_{n\ge 1}$. By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element $\varphi $ of the set ${\mathcal{P}}^{\u266e}\left(G\right)$ of $K$-biinvariant continuous functions of positive type on $G$.