TITLE: Why does a C.D.F need to be right-continuous?
QUESTION [5 upvotes]: As you may know, if $(\Omega,\mathcal{F},\mathbb{P})$ is a probability space and $X\colon\Omega\to\mathbb{R}^k$ is a random variable, then the cumulative distribution function of $X$ is defined as 
$$F_X(a):=\mathbb{P}(\{\omega\in\Omega\,\colon X(\omega)\leq a\})=\mathbb{P}\Big(X^{-1}\Big(\prod_{i\in [k]}(-\infty,a_i]\Big)\Big),\,\text{ for each } a\in\mathbb{R}^k.$$
This function is always right-continuous. That is, for each $x\in\mathbb{R}^k$ we have $\lim_{a\downarrow x}F_X(a)=F_X(x)$. 
My question is: Why is this property important? Is there any capital result in probability theory that depends on it?

REPLY [3 votes]: It doesn't "have" to be. A distribution function is defined either as 
$$F_X(x)=\mathbb{P}_X((-\infty,x])=\mathbb{P}(X\leq x)$$
Then it is right continuous (follows from continuity of measures from above). It could be defined as
$$F_X(x)=\mathbb{P}_X((-\infty,x))=\mathbb{P}(X<x)=1-\mathbb{P}(X\geq x)$$
Then it is left continuous, which again follows from continuity of measures.