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?