On some definition of expectation of random element in metric space

We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results: conditions under which the metric space is isometric with some real Banach space; conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.