let $\displaystyle \mu, \nu$ be two probability measures on compact hausdorff spaces X,Y. a measure $\displaystyle \eta$ on $\displaystyle X \times Y$ is a joining of $\displaystyle \mu, \nu$ if $\displaystyle \eta(A\times Y)= \mu(A),\; \eta(X\times B)=\nu(B)$. The set of all the joinings of these two measures is a convex subset of all the measures on $\displaystyle X\times Y$.

what I need to show is that the functions of type f(x)+g(y) where $\displaystyle f\in L^1 (X),\;g\in L^1(Y)$ are dense in $\displaystyle L^1(X\times Y,\eta)$ iff $\displaystyle \eta$ 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