let \mu, \nu be two probability measures on compact hausdorff spaces X,Y. a measure \eta on X \times Y is a joining of \mu, \nu if \eta(A\times Y)= \mu(A),\; \eta(X\times B)=\nu(B). The set of all the joinings of these two measures is a convex subset of all the measures on X\times Y.
what I need to show is that the functions of type f(x)+g(y) where f\in L^1 (X),\;g\in L^1(Y) are dense in L^1(X\times Y,\eta) iff \eta 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