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Prove that the metrics $\rho(A,B)$ defined by : $$ \rho(A,B) = \begin{cases} \dfrac{\mathbb P(A \bigtriangleup B)}{\mathbb P(A \bigcup B)} & \text{if} \space \mathbb P(A \bigcup B) \neq 0 \\ 0 & \text{if} \space \mathbb P(A \bigcup B) = 0 \\ \end{cases} $$ satisfy the triangular inequality.

What i was thinking was to use the fact that $ x \rightarrow \frac{a+x}{b+x}$ when a < b and $x \geq -a$ is increasing to swith from any C to subevent of $A \bigcup B$ and then normalizing the probability function, assume that $\mathbb P(A \bigcup B) = 1 $ but i don't know how to proceed.

I'm desperate in finding a solution.

Ricter
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  • As you are blocked on the general case, have you attempted to establish it for the simple discrete probability distribution, U([1,n]) (uniform on $n$ elements) : it amounts to counting a ratio of number of elements... – Jean Marie Oct 27 '22 at 21:15
  • Honestly I have no idea what you're talking about. Instead, I was thinking about the application of the Jaccard distance on measures. What do you think? – Ricter Oct 27 '22 at 21:44
  • I was tempted to give you this advice. I had written some time ago a question mentionning the definite positiveness of a certain matrix $A$ https://math.stackexchange.com/q/3173596/305862 in connection with Jaccard index. Maybe the property $\det(A)>0$ for the case $3 \times 3$ is equivalent to your looked for triangular property. – Jean Marie Oct 27 '22 at 21:58
  • I saw it before, but i couldn't get anything useful for me, but surely for my bad. – Ricter Oct 27 '22 at 21:59
  • "honestly, I have no idea what you are talking about". Maybe something is missing in your question leading to this misunderstanding of mine: you are working on subsets of $\mathbb{R}$ (more exactly borelian subsets) and not on a general probability space... – Jean Marie Oct 27 '22 at 22:02
  • This were the only conditions that the professor gave us, so no further info's – Ricter Oct 27 '22 at 22:05

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