A necessary and sufficient condition for probability measures dominated by gg-expectation

We prove that if the generator gg of a gg-expectation EgEg is independent of yy, then a probability measure QQ is dominated by EgEg if and only if QQ can be generated by the Girsanov transformation via a controlled process; furthermore, we prove that EgEg equals the supremum of its dominated mathematical expectations if and only if gg is sublinear with respect to zz.