I saw in a paper an argument, that seems simple. (Let $\triangle$ be the triangle operator of symmetric set difference between two sets) It states that if $P(A \triangle B) < \epsilon$, for some small $\epsilon>0$, then if $x \in A$ then it is very likely to be in $B$ and vice versa.
I am trying to figure out how we would formalize this notion. More specifically, I am trying to formalize the idea that if $P(A \triangle B) < \epsilon$ then $P(A) \approx P(B)$, or that we can use $A$ and $B$ interchangeably in a statement. Is it true, for example, that if $P(A \triangle B) < \epsilon$, then $|P(A)-P(B)|<\epsilon$?
I know that $P(A \triangle B)$ defines a pseudometric, so the question is really whether this pseudometric is dominated by the divergence measure of $|P(A)-P(B)|$. Any other ideas how to formalize this idea that set difference is small means sets are interchangeable would be great.
