Question about mixed strategies in all-pay auctions

Question

The following excerpt is taken from Martin J. Osborne's book on game theory. When considering a mixed strategy Nash equilibrium under a continuous random variable, why do we consider actions with zero probability? Here, $F(0)$ represents the probability of playing an action less than or equal to zero but since $a_i>=0$, this is the probability of playing an action equal to zero.

I thought that the definition of a mixed strategy specifies that expected payoffs must be equal only under actions played with positive probability so why does this work?

Answer 1

A mixed strategy given by a continuous random variable is represented by the random variable's probability density function or by its associated cumulative distribution function, here $F$. While every single action $a_i$ is played with probability zero, it can nevertheless have a positive probability density. In a mixed Nash equilibrium, a player's expected payoffs must be equal for all actions with positive probability density.