"Distribution functions should have disappeared long ago"

In "Mathematical Foundations of the Calculus of Probability" by Jacques Neveu, the author says the following about distribution functions:

These functions, which are in fact of very little practical use (except in certain questions where the order structure of the real line plays a predominant role), should have disappeared a long time ago to the benefit of the ensemble definition of the notion of probability.

This sentiment (that they should have disappeared long ago) is also expressed by Erhan Ҫinlar in "Probability and Stochastics".

I have two closely related questions:

• What is the justification for this view? I don't really see how you would come to this conclusion. How are distribution functions impractical?
• Is this a commonly held sentiment among experienced mathematicians?

For the record: The distribution function of a RV $X$ is the function $x\mapsto \mu(-\infty\,..x]$, where $\mu$ is the distribution of $X$.

I (think) the argument being made is how distribution functions only give you information about the probabilities of sets of the form $$X^{-1}( (-\infty, a])\,,$$ whereas the real source of interest is the values of the probability measure for sets of the form $$X^{-1}(A)$$ for any (Borel)-measurable set $A$.

I base this inference on the parenthetical comment from your quotation:

except in certain questions where the order structure of the real line plays a predominant role

In those certain cases I imagine that sets of the form $(-\infty, a]$ (and their inverse images under a random variable $X$) are more important than arbitrary Borel-measurable sets $A$.

This is, however, just a guess -- although I read Neveu's book a year or so ago, I don't quite remember what else (if anything) he mentioned about the topic. I have not yet read Cinlar's book, although I hope to do so in the future.