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?