let be two probability measures on compact hausdorff spaces X,Y. a measure on is a joining of if . The set of all the joinings of these two measures is a convex subset of all the measures on .

what I need to show is that the functions of type f(x)+g(y) where are dense in iff is an extreme point.

The only thing I managed to do is to prove this when X and Y are finite,which is a long way from the general case, so any idea on how to approach this will be welcome