Probability zero vs impossible I understand that probability $0$ does not mean 'impossible' - because if we look for instance at a uniform distribution over $[0, 1]$ then while each of the singleton events $\{r\}$ for $0\leq r \leq 1$ has probability $0$, if we carried out the experiment then we would get exactly one of the numbers in $[0,1]$, and so these events are not actually impossible, even though they have probability $0$.
However, if we look at a distribution defined by a density function which is zero on $[0,\frac{1}{2}]$ and non-zero on $(\frac{1}{2},1]$ (let's say with a continuous transition between them), then we know that for any $0\leq r\leq \frac{1}{2}$ the event $\{r\}$ is impossible, and for $\frac{1}{2}<r\leq 1$ the event is possible, yet still has probability $0$.
Both have probability $0$, but one of them is possible, and the other is not. Is there a definition that captures this distinction between the two cases?

As mentioned in other answers, one major sticking point is in the mathematical definition of "impossible". 
If you view a random variable as a measurable function
$$
\begin{align*}
X: \Omega &\to [0, 1] \\
\omega &\mapsto X(\omega)
\end{align*}
$$
taking elements $\omega$ of some sample space $\Omega$ (i.e. events) to a real number $X(\omega)$ (the probability of that event happening), one reasonable definition is 
$$
\text{An event } \omega \text{ is } \textbf{possible} \iff \omega \in \Omega
$$
Note that you can still have $X(\omega) = 0$, i.e. $\omega$ occurs with probability zero. Impossible events are then just those events not in the domain of $X$.
