What does the σ in σ-algebra stand for?

2011-08-29T21:27:27Z

I was tutoring someone in analysis and realized I have no idea where this notation comes from (or analogous terms: σ-additive, σ-ring, etc). I would like to know why the letter σ was chosen. I can't think of anything relevant that starts with "S" in either English or French. My German is nearly nonexistent, but I didn't see an explanation while trying to read the German wikipedia page.

Bonus points if you can tell me who introduced this notation and when.

(By the way, I really don't like this notation very much. I think it would be much more reasonable if we just wrote "$\aleph_1$-algebra" instead. Or better yet, replaced "algebra" with a less overloaded word. But I might change my mind, if it turns out there is a good explanation for the σ!)

Answer by Georges Elencwajg for What does the σ in σ-algebra stand for?

2011-08-29T23:42:54Z

From Elstrodt's book Maß- und Integrationstheorie, pages 13-14:

Bei den Wörtern „$\sigma$-Ring", „$\sigma$-Algebra" weist der Vorsatz „$\sigma$-..." darauf hin, daß das betr. Mengensystem abgeschlossen ist bez. der Bildung abzählbarer Vereinigungen. Dabei soll der Buchstabe $\sigma$ an „Summe" erinnern; früher bezeichnete man die Vereinigung zweier Mengen als ihre Summe (s. z.B. F. Hausdorff 1, S. 5 und S. 23).
Eine entsprechende Terminologie ist üblich mit dem Vorsatz „$\delta$..." für abzählbare Durchschnitte (z.B.„$\delta$ -Ring").

My translation:

In the words "$\sigma $-ring","$\sigma$-algebra" the prefix "$\sigma$-..." indicates that the system of sets considered is closed with respect to the formation of denumerable unions. Here the letter $\sigma$ is to remind one of "Summe"[sum]; earlier one refered to the union of two sets as their sum (see for example F. Hausdorff 1, p. 5 and p. 23).
A corresponding terminology is usual with the prefix „$\delta$-..." for denumerable intersections [Durchschnitte] (for example "$\delta$ -ring")

(The reference is to Hausdorff's Grundzüge der Mengenlehre. published in 1914.)

To sum up: the excerpt says that $\sigma$ [=Greek s] and $\delta$[=Greek d] come from the German words Summe and Durchschnitt, whose English translations are respectively sum and intersection.

What does the σ in σ-algebra stand for? - MathOverflow

What does the σ in σ-algebra stand for?

Answer by Georges Elencwajg for What does the σ in σ-algebra stand for?