You'll note that the notes don't derive the calculation of p-value. It merely gives a trivial example (curiously, the sum of two probabilities) with a fiat interpretation.

It's curious that my last post, merely quoting the third axiom and showing its properties plain, makes you think that I am deeply confused, when you cannot even refute trivial points and must resort to arguments (poor, in that they do not address our issue here) from authority.

> We will restrict ourselves to discrete sample spaces for now, to avoid some technical difficulties…

In the later lectures they take a look at sample spaces that are uncountably infinite. Good luck working with a sum there.

What reformulation of the probability axioms are you using to eliminate the sum in the third axiom?

The axioms may be trivially extended to bounded intervals (a la calculus).