I am learning about probability and the definition of pairwise independence is given as $P(AB) = P(A)P(B)$. My textbook motivates this definition as one to capture the intuition where the knowledge of one event occurring doesn't change the your knowledge of the likelihood of the other event occurring. I understand that this definition implies the two conditional probability statements that are closer to that above intuition, given that both events in question have non-zero probability.

But it seems to me that there are examples where this definition does not match intuition, when one or more of the events have zero probability. Consider an experiment where you pick a random number $R$ uniformly between 0 and 1. The event where that random number $R$ is equal to 0.5 has probability 0. So, by the standard definition of independence, it is trivially independent with the event that your number $R$ is in $[0.25, 0.75]$. But intuitively, the knowledge of $R = 0.5$ occurring should make you 100% confident that $R \in [0.25, 0.75]$.

I guess my question is why is independence defined for zero-probability events when the definition does not capture the intuition of independence? What is the utility of it?

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