Consider a family of $\cal{F}$ of subsets of a set $X$ and a function $\mu:\cal{F}\to[0,1]$. Further assume that $\mu$ either is defined or can be extended on the intersections $A\cap B$ for $A, B\in\cal{F}$ such that it is consistent with $\mu(A\cup B) = \mu(A) + \mu(B) - \mu(A\cap B)$.
A $\sigma$-algebra, $\sigma(\cal{F})$, can be generated from $\cal{F}$ somewhat constructively by a countable number of complement, union and intersection operations.
Question: Can we say that $\mu$ can be extended "alongside" this construction to the whole of $\sigma(\cal{F})$? If not, what is missing or how does it fail? Does $X$ have to be countable, for example, or $\cal{F}$?