Independent complements of subalgebras and product-measures

For a complete Boolean algebra $\mathcal X$ conditions are given on the structure of subalgebras $\mathcal X^1$ and $\mathcal X^2$ which guarantee the existence on $\mathcal X$ of a strictly positive measure $\nu=\mu^1\times\mu^2$ such that the marginal measures $\mu^i$ are the projections of $\mu$ to $\mathcal X^i$.