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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!)

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I assume it's the same $\sigma$ as in $F_\sigma$ sets, and probably stands for "somme" in French. However, that's just a guess. I agree that it's not great notation. –  Henry Cohn Aug 29 '11 at 21:35
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I believe it is actually not the same as in $F_\sigma$-set. The $\sigma$ in the context of $\sigma$-algebra, $\sigma$-additive, $\sigma$-closed always means "countable", while the $\sigma$ in $F_\sigma$ (for "somme" or "Summe" in German) is paired with $\delta$ in $G_\delta$ which I (as native German speaker) always think of as standing for "Durchschnitt" (intersection). –  Stefan Geschke Aug 29 '11 at 21:46
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Hmm, you could be right, but it's a countable union in both cases: an $F_\sigma$ set is a countable union of closed sets and a $\sigma$-algebra is closed under countable unions. Either way, the notation focuses on the union rather than the countability. As for German vs. French, I've always thought of $F_\sigma$ as French (fermé somme) and $G_\delta$ as German (Gebiet Durchschnitt). –  Henry Cohn Aug 29 '11 at 21:51
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Couldn't the $\sigma$ have originally come from the word for sum (which, in latin, is summa)? In the end, $\sigma$-algebras are designed to produce sets which play well with summation (capital $\Sigma$). There is the additional connection that the sign for the integral also comes from the latin word summa (and is supposed to be an elongated s), and $\sigma$-algebras are used to generate the modern integral. –  Peter Luthy Aug 29 '11 at 23:27
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Interesting all the weird guesses posted by those who don't know the answer! –  Gerald Edgar Aug 30 '11 at 0:10

1 Answer 1

up vote 34 down vote accepted

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.

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And of course "intersection" ("Durchschnitt") begins with D which corresponds to Greek $\delta$. Just as "union" ("Summe") begins with S which corresponds to Greek $\sigma$. –  Gerald Edgar Aug 30 '11 at 0:08
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For those with institutional access: Hausdorff's Mengenlehre (in the 1935 edition) is available electronically on SpringerLink: dx.doi.org/10.1007/978-3-540-76807-4_2 and I'd like to point to the page 136 where $G_\delta$ and $F_\sigma$ sets are introduced, here's a picture of that page: i.stack.imgur.com/NKWD1.png (note that $\aleph$ is the cardinality of the continuum). –  Theo Buehler Aug 30 '11 at 1:30
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Is Hausdorff's 1914 book the first place the term $\sigma$-algebra (or ring) appears? At first I was skeptical about the Elstrodt reference, since for example Dudley's book Real Analysis and Probability gives a different explanation for $F_\sigma$. However, I looked around and can't find any earlier source than Hausdorff. Lebesgue and Borel certainly had some version of the concept 10-15 years before Hausdorff's book, but I looked up some of their papers and they don't seem to formalize it this way or define a $\sigma$-algebra at all. Does anyone know whether Hausdorff was the first? –  Henry Cohn Aug 30 '11 at 3:37
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@Theo:thank you for the reference, and especially for the picture. Great (night?)work! By the way, how do you extract and display such pictures? (I'm asking publicly in case some other user as ignorant as I is interested, but if you prefer you can e-mail me the answer) –  Georges Elencwajg Aug 30 '11 at 7:44
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Nightwork indeed! The procedure for uploading a picture is not difficult but somewhat lengthy to describe: First you need a .png-file of the picture you want to upload. You can obtain this by making a screen shot, for example. To upload the picture I find it most convenient to (ab)use the stackexchange-site by opening an answer field, hitting ctrl-G and following the instructions there. You then get a link to the picture. This way the picture is guaranteed to be permanent. It may not be entirely proper to do it this way, but given that FogCreek is hosting mathoverflow I'm not having qualms. –  Theo Buehler Aug 30 '11 at 9:02

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