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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 $X$.

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    What does the author mean by the ensemble definition? – 2017-02-03
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    @littleO The term "ensemble definition" seems to occur only this one time in the book. Judging from the material before this quotation, I think he simply means "definition of probability" (as in "normed finite measure") since he procedes to show, that there is bijection between probability measures on $(\mathbb R, \mathcal B(\mathbb R))$ and distribution functions (at this point random variables have not been introduced yet). – 2017-02-03
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    I wonder how this person feels about PDFs? If distributions and PDFs disappeared then most people would never learn probability. – 2017-02-04
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    @Michael You are misinterpreting the statement. The books listed are rigorous books on measure theory and probability. No argument is made against distributions or densities. Only against the function $x\mapsto \mu(-\infty\,..x]$, where $\mu$ is the distribution of a given RV. – 2017-02-04
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    @StefanPerko : I have no idea what your sentence means, or what "For the record: The distribution function of a RV $X$ is the function $x \rightarrow \mu (-\infty, x]$ where $\mu$ is the distribution of $X$." Sounds like you are using the word distribution to define itself. The conventional definition of the distribution is the cumulative distribution function $F_X(x) = P[X\leq x]$. – 2017-02-05
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    @Michael Maybe in a first course in probability, but not beyond that. The conventional definition of distribution is the pushforward measure $\mathbb P \circ X^{-1}$ of the the ambient probability measure $\mathbb P$ along $X$. – 2017-02-05
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    Not an answer, but the [review of this book](https://www.jstor.org/stable/2027432?seq=3) by F. Eicker for *SIAM Review* mentions, and comments on, this particular passage. – 2017-02-06
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    @StefanPerko : Sounds like you are making a distinction between the terms “distribution” and “distribution function.” Not all books do that. I notice that David Williams avoids confusion in _Probability with Martingales_ by using terminology “distribution function” for $F_X(x) = P[X\leq x]$ and “law” for $L_X:\mathcal{B}\rightarrow \mathbb{R}$, $L_X = P \circ X^{-1}$. Since the law can be obtained from $F_X(x)$, it makes sense to emphasize the simpler $F_X(x)$ (usually math favors simplicity). Also, $F_X(x)$ can be plotted and seems essential for learning. – 2017-02-07
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    @Michael: in what was written before in Stefan Perko's comment, the distribution is $\mu = \mathbb{P}\circ X^{-1}$ and the distribution function is $F_X\colon x\mapsto \mu(-\infty,x]$. There is no confusion, and the two are distinct indeed (and it's also equivalent to what you wrote). – 2017-02-07
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    @StefanPerko "Distribution functions on the real line are mentioned in this chapter only incidentally, and are not used later on. Consequently characteristic functions or the central limit problem are not treated at all in this book. The author indicates his reasons for this as follows: *[your quote]* This again underlines the emphasis on measure theoretic aspects predominant in the book as can, of course, already be seen from the table of contents. On account of this, a slightly more restrictive title of the book would have appeared more appropriate to the reviewer." (review, p.138) – 2017-02-07
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    @ClementC. Thanks for that! Although I find this confusing (of course it is not your statement, but:). Do you happen to know what is meant by "Consequently" in the second sentence? Afaik, characteristic functions are a concept completely independent from distribution functions (Ҫinlar also speaks of the "Fourier transform" of the distribution). – 2017-02-07
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    @Michael I'd say math favors *abstraction* over simplicity. Measures are arguably way nicer, because they don't *need* to be defined on the reals and things like Radon-Nikodym work regardless of the underlying measure space. – 2017-02-07
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    @StefanPerko Your guess is as good as mine, but I gather the reviewer means that convergence in distribution (which corresponds to pointwise convergence of the distributions functions) is barely touched on: this explains the reference to central limit theorems (typically, convergence in distribution) and characteristic functions (whose convergence pointwise is equivalent to convergence in distribution). – 2017-02-07

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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.

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    Mmh. But these intervals generate the Borel-$\sigma$-algebra and the distribution function uniquely determines the distribution (finite measure specified on generating $\pi$-system), so I don't really "get" it. – 2017-02-03