If P(A)=0, is A a null event? I know that P(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 probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a probability measure on $\Sigma$.
One of the properties of $\Sigma$ is that it contains the empty set $\emptyset$, and the measure of this set must be zero. So the empty set has measure zero $P(\emptyset)=0$.
However there exist non-empty sets that also have a zero measure. E.g for the normal random variable the measure of a singleton ( which is  obviously  not empty) {a} is $\int_a^a \varphi(x)dx=0$. 
Consequently, the measure of a union of singletons, being the sum of the measures of each singleton in the union, is also zero.

So the  null event is the empty set that is element of $\Sigma$ and
  $P(\emptyset)=0$, but $\Sigma$ can contain non-empty sets of probability
  zero.

This gives rise to concepts like ''almost everywhere''; a property holds almost everywhere if it holds everywhere except on a set that has probability zero. 
A: First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As the first interpretation makes the question a tautology (of course if the definition of null event is that it's probability is zero, then a null event has zero probability and an event of zero probability is a null event), I'll concentrate on the second interpretation.
In the usual measure theoretic formulation of probability, ''event'' is a set of outcomes; an event is realized if the outcome of the experiment is within the set. Impossible event is the empty set $\emptyset$, i.e. under no outcome of the experiment can this event be realized.
The answer to your question is no. Let $X$ be a random variable with uniform distribution on $\left[0,1\right]$ and $A$ be the event $X=0.5$ (or any other real number on $\left[0,1\right]$). This is obviously not a null event (such random variate can take the value of $0.5$) but has the probability of zero (as the distribution is continuous).
Another example might be having infinite number of heads when flipping a fair coin. (''Infinite number'' might be formalized, but I don't want to make the discussion too technical, consider it intuitively.) This can happen (that is: the event pertaining to it is not an empty set), yet, its probability is zero.
See also this discussion.
A: The idea of a Null event is used to emulate the idea of a failed experiment. 
Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.
First you have the probability that you did, in fact flip a coin. This has a probability of 1 technically speaking. 
The second is the Null event (usually denoted with a probability of 0). This null event means the experiment failed. Because you know you flipped a coin with a probability of 1 the only real null events for this experiment involve not being able to read the coin. Maybe you flipped it and it rolled into a crack in the floor, and you were unable to get conclusive results. The null event is not a number used in the math, but models the finite improbability that you cannot complete the experiment. You can think of it as 0+ (if you are familiar with limits/asymptotic expressions) because there aren't a lot of absolutes in statistical analysis. 
Third and fourth are the probability that you flipped heads or tails. Both of these are considered to be .5 or 1/2 because there are only two real options. 
The null event and the first are used to tell whether the experiment worked or not. Something with a probability of 0 is not necessarily a null event. Take for example the probability that you will draw a red marble from a bag of blue marbles. This has a probability of 0, but is not a null event because you did in fact draw a marble.  
Hopefully that clarifies the Null event for you. 
