Let $\mathcal{A_1}$ and $\mathcal{A_2}$ be $\sigma$-algebras of subsets of some space X. Suppose $\mu_j$ is probabilistic measure on $\mathcal{A}_j$ for $j=1,2$.

**Question:** What are the necessary and sufficient conditions for existence of common extension of these measures to some probabilistic measure on $\sigma(\mathcal{A}_1,\mathcal{A}_2)$?

The obvious necessary condition is as follows: $\forall U_i\in \mathcal{A_i}$, $i=1,2$, if $U_1 \subset U_2$ then $\mu_1(U_1)\leq \mu_2(U_2)$ and vice versa. It is known that if we are interested in finitely-additive measures then this condition is sufficient.

What about general $\sigma-$additive measures?