Continuity of Probability if $An\to A$ but neither increasing or decreasing Perhaps this is a silly question, but most books of probability talk about continuity of probability if either $A_n$ increases to $A$ or if $A_n$ decreases to $A$. My question is, if we just have that $A_n \to A$, does continuity of probability still holds? If so, why not just state that instead of separating the increasing and decreasing cases?

The answer by @Momo proved that a probability measure is indeed continuous. Now, my other question was why not presenting such property instead of separating continuity from above and from below. I think I’ve figured why most books do this. The reason would be that continuity from above (or below) plus finite additivity is equivalent to $\sigma$-additivity. Hence why won would prove them instead of the stronger claim of continuity.
