Rationale behind defining distribution function with strict inequality A friend from Italy pointed me to a nice Probability Theory book written in Italian by G. Dall'Aglio, a well-known Italian probabilist; Original title: Calcolo delle Probabilità, terza edizione, Zanichelli, ISBN 978-88-08-17676-9.
I discovered that the author defines the cumulative distribution function of a random variable $X$ as
$$
F(x) = P(X<x), \tag{*}\label a
$$
i.e. using strict inequality instead of the usual $\leq$.
In this unusual definition (I am not aware of other scholars defining the distribution function this way), the $F(x)$ turns out to be left-continuous, though the other properties are the same as those of the usual distribution function. The book, as far as I can tell, fails to motivate such an unusual choice.
Is there a reason to prefer $\eqref a$ to the usual definition or, is the choice between the two definitions a matter of taste?

Like others have mentioned the use of $<$ instead of $\leq$ is just a convention. Here I want to post a remark which is, I believe, interesting enoughy to be placed as an answer.
The convention $<$ was used by Kolmogorov in 'Foundations of the Theory of Probability' (see page 15 of this 1956 second edition of the English translation).

The function $$F^{(x)}(a) = P^{(x)}(-\infty,a) = P \{ x<a \}$$ where $-\infty$ and $\infty$ are allowable values of $a$ is called the distribution function of the randome variable $x$.

It is remarkable that he uses the interval $(-\infty,a)$ instead of $[-\infty,a)$. Maybe Kolmogorov used the $<$ convention, instead of the $\leq$ convention, because he didn't want to use an interval with infinity inclusive.
If $a = \infty$ then  $F^{(x)}(a) = P^{(x)}(-\infty,\infty)$ and using $P^{(x)}(-\infty,\infty]$ instead would be problematic.
I believe this might be a reason for Kolmogorov's convention, although I would not understand exactly why there would be a problem with including infinity in the interval. Maybe because it is not an element in the sample space?
