Let $(X,\mathscr{F})$ be a measurable space.

Let $\mathscr{F}_\alpha \subseteq \mathscr{F}$, $\alpha \in \mathfrak{A}$ be a class of $\sigma$-subalgebras.

Let we have a measure $\mu_\alpha$ on every $(X,\mathscr{F}_\alpha)$.

Let the family $(\mu_\alpha, \alpha \in \mathfrak{A})$ be consistent, i.e. 
$\mu_{\alpha}(A)\le\mu_{\beta}(B)$ when $A\subseteq B$.

Now the question is if it is possible to construct such a measure $\mu$ on $(X,\mathscr{F})$ that $\mu(A)=\mu_{\alpha}(A)$ for all $A \in \mathscr{F}_\alpha$, $\alpha \in \mathfrak{A}$ ?