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I am reading Statistical Theory: A Concise Introduction, by Felix Abramovich and Ya'acov Ritov. In the the appendix, the authors provide a primer on basic probability theory. In discussing the probability function, the authors write the following:

The probability function assigns to each event $A \in \mathcal{A}$ a real number $P(A)$ called the probability of $A$ satisfying the following conditions:

...

  1. If $A_1, A_2, ...$ are disjoint events, that is, at most only one of them can happen and $P(A_i \cap A_j) = \emptyset$ for all $i \not= j$, then $P(A_1 \cup A_2 \cup \ ...) = \sum_{i = 1}^\infty P(A_i)$.

$A$ denotes an event, and $\mathcal{A}$ is a set of events.

My question is, shouldn't it be $P(A_i \cap A_j) = 0$? After all, it makes no sense to say that $P(A_i \cap A_j) = \emptyset$.

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It should indeed be either $A_i\cap A_j=\varnothing$ or $P(A_i \cap A_j) = 0$. Apparently the author(s) accidently mixed them both up and the editor didn't notice this error.

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