Equivalent or absolutely continuous probability measures with given marginals

Let $(mathcal{X},mathcal{A})$ and $(mathcal{Y},mathcal{B})$ be measurable spaces. Suppose we are given a probability $alpha$ on $mathcal{A}$, a probability $eta$ on $mathcal{B}$ and a probability $mu$ on the product $sigma$-field $mathcal{A}otimesmathcal{B}$. Is there a probability $ u$ on $mathcal{A}otimesmathcal{B}$, with marginals $alpha$ and $eta$, such that $ ullmu$ or $ usimmu$ ? Such a $ u$, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of $ u$ are provided, distinguishing $ ullmu$ from $ usimmu$.