THE SET OF MEASURES GIVEN BY BOUNDED SOLUTIONS OF THE COMPLEX MONGE–AMP`ERE EQUATION ON COMPACT K¨AHLER MANIFOLDS

Consider the image of the Monge–Amp`ere operator acting on bounded functions, defined on a compact K¨ahler manifold, whose sum with the local K¨ahler potential is plurisubharmonic. It is shown that a nonnegative Borel measure belongs to this image if and only if it belongs to the image locally. In particular, those measures form a convex set.