Why are half-open intervals $(a,b]$ "special" in probability theory? I'm learning probability theory and I see the half-open intervals $(a,b]$ appear many times. One of theorems about Borel $\sigma$-algebra is that

The Borel $\sigma$-algebra of ${\mathbb R}$ is generated by inervals of the form $(-\infty,a]$, where $a\in{\mathbb Q}$. 

Also, the distribution function induced by a probability $P$ on $({\mathbb R},{\mathcal B})$ is defined as
$$
F(x)=P((-\infty,x])
$$
Is it because for some theoretical convenience that the half-open intervals are used often in probability theory or are they of special interest?
 A: The fundamentally nice properties of half-open intervals are that:


*

*They are closed under arbitrary intersections

*For two half-open intervals $I_1, I_2$, their difference $I_1 \setminus I_2$ is a union of half-open intervals (a trivial union for $\Bbb R$, but not so in $\Bbb R^n$, in general)


That is, these half-open intervals form a so-called semiring of sets.
This is important because Carathéodory's theorem (on existence of measures) grips on such semirings; this route then leads to the theorem that Lebesgue measure on $\Bbb R^n$ exists.
I think this is one of the main reasons why probability and measure theorists like this type of interval.
A: I think it's because the distribution function in the discrete case is the sum of probabilities from minus infinity up to and including x; but minus infinity is not a number so that end of the interval is open, i.e., has no end point.
A: The half-open intervals are not necessarily special in a particular way, they are one of many possible generators of the Borel $\sigma$-algebra. 
As I understand it, most of the things you do with half-open intervals you could also do with other generators, but in practice they are easy to work with
A: I would say that the answer is best presented in the reverse order of your question:
The (cumulative) distribution function is defined the way it is, because that is a natural way to think about accumulating probabilities as you observe more events in the sample space. So in a sense, the half-open intervals are the most fundamental events to consider. This is, in essence, what Hagen mentioned.
Also, sigma-algebras define what combinations of sets (i.e. the fundamental events) are also considered as events and thus can be assigned probability. The set of all such events on the real line, generated by the half-open intervals are the Borel sets.
