Example where $E \subset S$, $E \neq \emptyset$, and $P(E) = 0$? From what I understand, this situation isn't possible since any non-empty subset of S will have a non-zero probability. What am I missing here?
 A: This happens in the continuous case, for instance, where the probability of a specific value is null.
Let's consider the example of an arrow hitting a target, which is a disk of radius 1. Imagine it centered in the origin of the Cartesian plane.
Let S be the disk. So P(S)=1, as the arrow have to hit the target somewhere. If you consider for instance E={(1,1),(1,-1),(-1,1),(-1,-1)}, we have that $P(E)=0$, with $E \in S$ and $E \neq \emptyset$, as the probability for the arrow of hitting an exact point is $0$.
At the same time, there are elements in S that have a positive probability. For instance, the disk centered in the origin with radius $1/2$, let's name it S', have a positive probability. So, $S' \in S$, $S' \neq \emptyset$ and $P(S') \neq 0$. Actually, $P(S')=1/4$.
In the continuous case this happens due to a consistency problem. There are infinite points in an interval/disk/region, which we know having probability 1, or at least a finite probability. If we give a positive probability to each of the points, the total sum would diverge, as we would sum infinite fixed positive numbers.
