# Why is an ($\boldsymbol{\sigma}$-)algebra called a ($\boldsymbol{\sigma}$-)field in probability?

This is not strictly a math problem, but more of a question on nomenclature and history; so, if it's not appropriate for this site, let me know, and I'll ask someplace else.

Probability theory is known for its distinct nomenclature pattern that sets it apart from measure theory proper. Most of these names reflect the subject's informal beginnings and strong subsequent connections with mathematical statistics and other applications. I am listing as many of these differences as I can think off the top of my head:

measurable space --- sample space

measurable set --- event

almost everywhere --- almost surely (less commonly, almost certainly)

measurable function --- random variable

(Lebesgue) integral --- expected value

$\mathscr{L}_2$-norm --- standard deviation (not exactly, but similar)

$\mathscr{L}_2$-inner product --- covariance (again, not exactly)

weak-$\ast$ convergence --- convergence in distribution/law

($\sigma$-)algebra --- ($\sigma$-)field

Most of these probabilistic alternatives are more intuitive, except for the last one! I can't think of any empirical/pragmatic/intuitive/statistical reason to call a collection of 'events' either an 'algebra' or a 'field'. The name 'algebra', though, makes sense from a strictly mathematical point of view (the same way 'rings' are rings). So, why then do probabilists insist on 'field'?

Of course, tradition may be an answer, but I am more interested in knowing what/who started this tradition. Were 'algebra' and 'field' synonyms in the early days of Lebesgue theory, and over time, only one name survived in each of the two literatures? Do the two terms have to do with different mathematical schools, say, Moscow versus Paris, or something like that?

• In the french wikipedia there is a note saying that Kolmogorov historically used the term corps de Borel which literaly means Borel field. I suppose he said that in russian :) – Gribouillis Sep 16 '17 at 21:43
• @Gribouillis From what little I speak, I think an 'algebra' is 'алгебра' [ˈäɫgʲɪbɾə] even in Russian, though names might have been less standardized in Kolmogorov's time, and there might be alternative names, as well. Anyway, thanks for that piece of info. – sami.spricht.sprache Sep 17 '17 at 15:57
• The russian wikipedia article is not Kolmogorov's monograph Grundbegriffe des Wahrsheinlichtkeitsrechnung. I have an english translation here where the word field is used with a reference to Hausdorff. As the footnote seems to be from Kolmogorov, it means that he probably borrowed the term from Hausdorff. The translation is at kolmogorov.com – Gribouillis Sep 17 '17 at 16:39
• @Gribouillis That does it, I suppose! I checked a copy of Hausdorff's Grundzüge der Mengenlehre for any reference to 'fields', and Hausdorff does call the structure we call an algebra, 'Körper'. Indeed, in a footnote, he explains (with warning): Die Ausdrücke Ring und Körper sind der Theorie der algebraischen Zahlen entnommen, auf Grund einer ungefären Analogie, an die man nicht zu weitgehende Ansprüche stellen möge. That does, however, leave the question Who came up with 'algebra'? open. – sami.spricht.sprache Sep 17 '17 at 23:34
• The word algebra is used in the general notion of field of sets. It may come from Boolean algebras in the 19th century. – Gribouillis Sep 18 '17 at 7:45