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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.

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closed as off-topic by Carlo Beenakker, Chris Godsil, Andrey Rekalo, Ricardo Andrade, David White Oct 4 '13 at 13:13

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1 Answer 1

This question may be more appropriate for Math StackExchange.

Nevertheless,

some hints:

For the argument from the paper, consider what would happen if $\mathbb P(A)$ and $\mathbb P(B)$ are close to $\epsilon$ as well.

For the other question: $$ A\triangle B = (A\setminus B)\cup (B\setminus A),$$ $$ A= (A\setminus B) \cup (A\cap B),\quad\text{and}$$ $$\mathbb P(X\cup Y)=\mathbb P(X)+\mathbb P(Y)\quad\text{if }X\cap Y=\emptyset.$$

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