I im trying to prove the following,
Suppose that $P$ is a probability measure on a field $\mathcal{F}$. If $A_t \in \mathcal{F}$ for $t>0$ with $A_s \subset A_t$ for $s<t$ and, $A=\bigcup_{t>0}A_t$, then $P(A_t)\uparrow P(A)$.
I'm having difficulties to prove this when $(A_t)_{t>0}$ is uncountable, as (I think) we cannot use the trick to split the union in a union of disjoint set and then use countable additivity.
Fix $\epsilon > 0$.
Consider the sequence of sets $A_1, A_2, A_3, \dots$. Verify that their union is $A$. So by countable additivity, their probabilities converge to $P(A)$; in particular there exists an integer $n$ so large that $P(A) - \epsilon \le P(A_n) \le P(A)$.
Now verify that for every real $t \ge n$ we have $P(A) - \epsilon \le P(A_t) \le P(A)$. This proves that the limit is as desired.