Why did Borel reject countable additivity of probability?

Question

On page 26 of the 2003 article - The origins and legacy of Kolmogorov's Grundbegriffe by Shafer and Vovk, it is stated that Borel rejected the countable additivity axiom of probability theory as he felt that a countable infinity of zero probabilities might add to a positive number. Was there a precedent for this in terms of an abstract or concrete application of probability theory in which this actually occurred? Or was his position simply being conservative about the axioms?

Answer 1

Even Kolmogorov himself "rejected" countable additivity, in the sense of not making it a universal property of probability. His first chapter axiomatizes and studies finitely additive probability measures, and countable additivity is only added in the second as an extra assumption. All he has to say about it in Foundations of the Theory of Probability (1933) is this:

"We limit ourselves arbitrarily to only those models that satisfy Axiom VI. This limitation has been found expedient in researches of most diverse sort".

The natural density on natural numbers is a classical counterexample that Kolmogorov was well aware of (it was featured in Khinchin's Zur additiven Zahlentheorie (1932) among many other places around 1930, see Niven, The Asymptotic Density of Sequences). The probability of uniformly choosing any particular natural number is $0$, but the probability of choosing at least one is, of course, $1$.

Borel did not reject countable additivity because of such examples, which everybody knew about, but because of suspicions about infinitary set-theoretic arguments that Lebesgue and others (Borel himself included, as it turned out, albeit unwittingly) introduced into measure theory. Borel was a proto-constructivist and platonistic tendencies of Cantor's set theory did not sit well with him. Accordingly, he preferred computation of probabilities as limits for philosophical reasons, because they clearly present the infinite as merely an idealization of the finite. Borel's seminal 1909 paper on the subject was titled Les Probabilities Denombrables et Leurs Applications Arithmetiques (although, as Kac quipped, "all of its theorems are true but almost all of the proofs are false"). Shafer and Vovk say as much:

"Borel’s discomfort with a measure-theoretic treatment can be attributed to his unwillingness to assume countable additivity for probability (Barone and Novikoff 1978, von Plato 1994). He saw no logical absurdity in a countably infinite number of zero probabilities adding to a nonzero probability, and so instead of general appeals to countable additivity, he preferred arguments that derive probabilities as limits as the number of trials increases (1909a, §I.4). Such arguments seemed to him stronger than formal appeals to countable additivity, for they exhibit the finitary pictures that are idealized by the infinitary pictures. But he saw even more fundamental problems in the idea that Lebesgue measure can model a random choice (von Plato 1994, pp. 36–56; Knobloch 2001). How can we choose a real number at random when most real numbers are not even definable in any constructive sense."

Ironically, Borel had no compunction about using what he called "countable independence", and considered it essential to his "denumerable probability theory". Ironically, because it is equivalent to countable additivity of the product measure, the very property he was so reluctant to employ, but on which his theorems depended. This lack of realization was responsible both for errors in his paper, and for overlooking proofs and results that would have made it stronger, see Barone-Novikoff, A history of the axiomatic formulation of probability from Borel to Kolmogorov:

"BOREL regarded countable independence as the essential, new (and probabilistic) ingredient of his new theory. By contrast he used countable additivity seldom (often surreptitiously), and he never explored its implications; about it he had reservations so deep that he frequently offered "alternative" proofs to evade reliance on it.

This assessment, which we shall defend by suitable analysis, explains at least in part why BOREL failed to draw the conclusion, attributed to CANTELLI (1917a, 1917b)... even if the trials are not assumed independent. BOREL's fascination with the principle of countable independence similarly may explain his failure to use anywhere in BOREL (1909) ... countable sub-additivity which follows from countable additivity even if the $B_i$ are not mutually incompatible."