# Were $\sigma$-algebras defined for probability?

If you want a crash course in $$\sigma$$-algebras and probability spaces, you should absolutely read this amazing answer by @Sycorax on Cross Validated. Sycorax says something in particular though that jumps out at me:

The three requirements of a $$\sigma$$-field can be considered as consequences of what we would like to do with probability: A $$\sigma$$-field is a set that has three properties:

1. Closure under countable unions.
2. Closure under countable intersections.
3. Closure under complements.

According to my literal interpretation, this suggests that the defining characteristics of a $$\sigma$$-algebra (which are usually phrased a bit differently it seems) were formulated with probability spaces in mind. Then again, the notion of something and its formal definition often come at different times.

Therefore, I have two closely related questions:

1. Was the notion of a $$\sigma$$-algebra devised with probability spaces in mind?
2. Were the points in the (widespread) definition of $$\sigma$$-algebras formulated with probability spaces in mind?

I haven’t found any online sources that reference the history of $$\sigma$$-algebras other than this explanation of the etymology of the $$\sigma$$.

– nwr
Commented Apr 23, 2020 at 18:37
• The MSE thread Origin of σ -algebras cites Émile Borel, 1898, "Leçons sur la théorie des fonctions : principes de la théorie des ensembles en vue des applications à la théorie des fonctions" as predating Lebesgue's work on measure theory.
– nwr
Commented Apr 23, 2020 at 18:41
• @Nick No hablo francés con fluidez :( Commented Apr 23, 2020 at 18:43

No and no, I am afraid. Those things are not specific to probability, we do them with logical connectives, which parallel set operations, and with areas and volumes, just as well as with probabilities. And countable additivity is not necessarily natural for some classical applications (e.g. to natural number frequencies), it is a technical assumption. Indeed, Kolmogorov's Foundations of the theory of probability (1933), that axiomatized probability in terms of measure theory, does not ask that the collection of sets be a $$\sigma$$-field in the initial five axioms, only a field, and that the probability measure be $$\sigma$$-additive, only finitely additive. Countable fields only appear in chapter II, and he formulates the last axiom in terms of measure continuity, not additivity.

The concept of set field he borrows from Hausdorff's Grundzüge der Mengenlehre (1914, English translation), which is not addressing probability at all. And the idea of considering collections of sets (or rather classes back then) closed under standard operations goes back to Boole and de Morgan, who were not even doing measure theory, they were interested in logic. Measure theory was originally developed for the case of real number line, with Peano-Jordan, Borel and Lebesgue introducing set functions on systems of sets, whose properties were eventually packaged into set fields, $$\sigma$$-fields, and $$\sigma$$-algebras.

Borel was the first to require $$\sigma$$-additivity in Leçons sur la théorie des fonctions (1898), but his interest in probability only develops around 1905. Lebesgue took a more abstract view of sets beyond subsets of a line in Leçons sur l'intégration et la recherche des fonctions primitives (1904), although he states his conditions in terms of integrals rather than measures. Neither Lebesgue nor Hausdorff were much involved with probability theory. A short review of this gradual measure-theoretic abstraction process is Le processus d'abstraction dans le développement des premières théories de la mesure by Villeneuve (it is in French but Google produces a decent translation).

For longer treatments see Classical and Modern Integration Theories by Pesin, Lebesgue's Theory of Integration: Its Origins and Development by Hawkins, and on connections to probability specifically Sources of Kolmogorov's Grundbegriffe by Shafer and Vovk, and Feller’s Early Work on Measure Theory and Mathematical Foundations of Probability by Fisher.

• FYI, in addition to Hausdorff's 1914 book, a historically significant paper (in the sense of its influence on later work, as was the case with Hausdorff's book) for putting the focus on abstract $\sigma$-algebra properties is Sur l'intégrale d'une fonctionnelle étendue à un ensemble abstrait by Fréchet (1915). Commented Apr 24, 2020 at 10:23
• By the way, the "English translation" of Hausdorff's book is an English translation of the 1935 3rd edition. The 1914 1st edition contains much material on measure theory and topology that was omitted in the latter two editions and was never translated into English (unfortunately). There is also a 1927 2nd edition (also never translated into English) in which much of the material on measure theory and topology was omitted and replaced with an expanded coverage of Borel and analytic sets (and related abstract set issues) proved in the intervening years by Lusin, Sierpinski, and others. Commented Apr 24, 2020 at 10:31
• @DaveLRenfro Thank you, I did not know. In the light of this I should mention that Kolmogorov specifically cites 1927 edition. Commented Apr 24, 2020 at 11:29
• The 2nd and 3rd editions also had a title change to Mengenlehre. (BTW, I have a hardback copy of the somewhat rare 2nd edition.) My notes on the 2nd edition: The 2nd edition contains an additional chapter, not present in the 1st edition, that includes topics such as the Baire property and negligibility of first category sets (and, more generally, sets from any $\sigma$-ideal), the Banach category theorem, localization relative to an ideal, and separation by Borel sets. However, the 2nd edition also omits many topics that were included in the 1914 1st edition, Commented Apr 24, 2020 at 11:41
• including: Lebesgue measure and integration, most of the results for general topological spaces, proof of the Jordan Curve Theorem, discussions on the paradoxes of set theory, and much of the material on ordered sets. My notes on the 3rd edition: The first 9 chapters of the 3rd edition are an almost unchanged reprint of the first 9 chapters that make up the 2nd edition, while the 10th (and last) chapter of the 3rd edition is new. The 10th chapter discusses the Baire property (for functions and for sets) and functions such that the inverse image of each point is an at most countable set. Commented Apr 24, 2020 at 11:41