Suppose that $X$ is a set, $J$ is a semi-ring with respect to $X$, $f$ is a pre-measure defined with respect to $J$, by Caratheodory's Extension Theorem there exist a measure $\mu$ such that $\mu$ agrees with $f$ on the elements of $J$. My question is:
If there exist a countable union of sets in $J$ ( a semi-ring ) such that their union is $X$ can be ensured that this extension is unique ?
In this proof:
https://sites.stat.washington.edu/jaw/COURSES/520s/521/HO.521/Caratheodory-unique.pdf
In justification (3) is used that if $v$ is the other measure then $v(X) = \mu(X)$, they agree on the set $X$, I don't see why this is true. In the proof of Rosenthal's book ,probabilities measures are used, so both values are equal to $1$, so how is this proved in the case that $\mu$ is a general measure, and not a specific probability measure.