If the probability of an event is $0$ , does it mean that the event is empty?

Question

If the probability of an event is $0$, does it mean that the event is empty?

Answer 1

Actually, no. For example if we define $\Pr([a,b])=b-a$, where $[a,b]\subset [0,1]$, then even though $\{0.5\}=[0.5,0.5]$ is not empty, the probability is $0$.

Answer 2

This is not even true on a discrete sample space. Consider $\Omega=\{1,2\}$ and define $p(\{1\})=0$ and $p(\{2\})=1$. Then $p(\{1\})=0$ but $\{1\}$ is nonempty.

Answer 3

Think of tossing a fair coin. You can make the sample space as $\{head, tail, edge\}$. And the probabilities are: $$P(H)=P(T)=0.5, P(E)=0.$$