On the existence of certain measures appearing in distance geometry

Abstract.

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure $\mu$ on X such that the value of¶¶$\int\limit _X f(x,y)d\mu (y)$ is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x ₁,x ₂,...,x _n,y ₁,y ₂,...,y _n in X, there exists an x in X such that¶¶$\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).$