If $ P(A) = 0 $ is $ A $ a null event? I know that $ P(\text{null event}) = 0 $, but is the reverse true? i.e. if $ P(A) = 0 $ is $ A $ a null event?
I'm not too sure I even understand what a null event is, to be honest. Could anyone give me an example of one?
 A: A null event is an event that is impossible.  Or more precisely, since an event is a subset of a sample space, the null event is the empty set.  So $P(\emptyset)=0$.  However, in some situations, a nonempty event may have probability $0$.
For instance if $X$ is a uniform $[0,1]$ random variable, then $P(X=\frac12)=0$, and yet $X=\frac12$ is not an empty event.
A: No.  
$P(A)=0$ merely means that the probability of event A is immesurable.  It does not necessarily mean that it is impossible.
Let $X$ be a random variable uniformly distributed over the interval $[0,1]$.  It will thus certainly not have a value outside that interval; it's impossible.  However, although it will have one value on that interval, the probability that it will have any exact value in that interval is almost surely zero; it can theoretically happen, but it's immesurable.
The distinction is sometimes denoted by the indicator a.s. (almost sure) or a.c. (almost certain), when you wish to be precise about why its measure zero.
$$P(X=2)=0$$
$$P(X=1/2)=0\quad\mbox{a.s.}$$ 
A: If there exists an event (A), such that P(A) = 0, A is a null-event.
Null event IS NOT the same as an impossible event. 
The best example to demonstrate this fact is the linear darts problem. Let's say, you throw a dart that falls anywhere within the interval [0,1]. 
$X$ $\in$ [0,1], where $X$ is the random variable that captures the position of the dart on the interval. 
We know, $P(X = 1/3)$ = 0. That doesn't imply the dart falling on position (1/3) is an impossible event! 
For every probabilistic experiment with a sample space $\Omega$, we can define a trivial $\sigma$-algebra that contains {$\phi$, $\Omega$}. $\phi$ is defined to be the impossible event and $P(X = \phi) = 0$.
Therefore, P(null-event) = 0 $\forall$ null-events; but P(A) = 0, implies A could either be a null-event or an impossible event($\phi$).
