I have found the following formal definition of a discrete random variable, $X$, taking values $x_1$, ..., with probabilities $p_1$, ... :
$F(x) = \mathbb{P}(X \leq x) = \sum_{i}p_i \mathbb{I}_{[x_i, \infty)}(x)$
I don't get the $\mathbb{I}_{[x_i, \infty)}(x)$ notation. I know it represents a set but I don't know what is its exact meaning here.
$[x_i;\infty)$ is the interval $\{x\in\Bbb R: x_i\leq x\}$ , the set of all real values that are $x_i$ or between $x_i$ and $\infty$.
$\Bbb I_A(x)$ is an indicator function, having value $1$ when the argument is in the set $A$, and $0$ elsewhere.
$$\Bbb I_{A}(x) =\begin{cases} 1 &:& x\in A\\ 0 &:& \text{otherwise}\end{cases}$$
In this particular case since $x\in[x_i;\infty)$ means $x_i\leq x$, then we have: $$\Bbb I_{[x_i;\infty)}(x) = \begin{cases} 1 &:& x_i\leq x\\0 &:& x< x_i\end{cases}$$
So, basically you are being told that the cummulative distribution function is the sum of $p_i$ values for all $i$ where $x_i\leq x$.
$$\sum_i p_i\Bbb I_{[x_i;\infty)}(x) = \sum_{i~:~x_i\leq x} p_i$$