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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 ${𝒫}^{♮}\left(G\right)$ of $K$-biinvariant continuous functions of positive type on $G$.