## joining of two measures

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