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The function $P$ that takes an event $A$ as input and returns the probability $P(A)$ as output is called a "probability measure" when we are developing probability using measure theory.

I have also seen some authors use the term "probability law". (I think Bertsekas uses this term.)

I think it is fairly common to refer to $P$ as a "probability distribution", but there is an issue here that I want to be careful about: I have seen that some authors use the term "distribution" only when referring to the distribution of a random variable $X$. For example, I believe that Folland makes this distinction. Should I make this distinction also in an undergraduate probability course (that does not use measure theory)?

What is the best or most standard term to use instead of "probability measure" in an introduction to probability course?

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  • $\begingroup$ A good piece of advice is to follow the terminology of the textbook we are using, but I'm also interested in understanding if my textbook is using the best terminology. $\endgroup$ – eternalGoldenBraid Jul 19 '18 at 16:01
  • $\begingroup$ Are you asking about a priori or a posteriori probabilities? $\endgroup$ – Jasper Jul 20 '18 at 2:35
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    $\begingroup$ I hope whoever answer this gives it 110%. $\endgroup$ – James S. Cook Jul 21 '18 at 23:37
  • $\begingroup$ @Jasper I'd like to know the best synonym for "probability measure" to use in an undergraduate introduction to probability course. It seems like a good idea to avoid using the word "measure" in a class that does not use measure theory. Perhaps "probability distribution" is the simple answer, but the terminology does not seem to be quite standard, as can be seen on this Wikipedia page. The textbook I'm using uses the term "probability set function", which I don't think is a very standard term. $\endgroup$ – eternalGoldenBraid Jul 22 '18 at 1:42
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    $\begingroup$ If you want to avoid saying "measure", speak of the "probability function" or simply the "probability". Another way to deal with this issue is to avoid speaking of $P$ in the abstract, always referring to $P(A)$ or $P(A|B)$, the probability of the event $A$ or the conditional probability of $A$ given $B$. $\endgroup$ – Dan Fox Jul 24 '18 at 9:52

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