Let F be a field over probability space §..the caratheodory extension criterion only extends this to a sigma algebra generated by F..i know that a larger sigma algebra cannot satisfy the countable additivity of lebesgue outer measure, but is there one which can satisfy the properties of probability function even if not that of lebesgue measure?
Let $m^*$ be Lebesgue outer measure on $[0,1]$. Let $A \subseteq [0,1]$ be a non-measurable set. That is, $m^*(A) + m^*([0,1]\setminus A) > 1$. Of course $A$ fails the Caratheodory criterion. But there is a countably-additive extension $\lambda$ of Lebesgue measure to a sigma-algebra including both the Lebesgue measurable sets and $A$.
Is that the question?