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?

A "conceptual consistency criterion" to choose between them per case, is when a value $x^*$ in the support is a "threshold", and then it matters whether consequences kick in if one reaches this threshold or if ones exceeds it.

*

*In the first case (consequences kick in when reaching the threshold), using $F_X(x) \equiv \Pr(X<x)$ appears to be the appropriate choice to reflect the conceptual separation of the two "states of the world," since $\Pr(X<x^*)$ describes the probability of "no consequences".

*In the second case (consequences kick in when exceeding the threshold), one would use $F_X(x) \equiv\Pr(X\leq x)$, which is the "no-consequences" situation in this case.

This is just conceptual for a continuous r.v., but becomes important when the random variable has jumps or is discrete.
