What does the σ in σ-algebra stand for? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:09:08Z http://mathoverflow.net/feeds/question/74004 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74004/what-does-the-in-algebra-stand-for What does the σ in σ-algebra stand for? Oliver 2011-08-29T21:27:27Z 2011-08-30T07:00:35Z <p>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.</p> <p>Bonus points if you can tell me who introduced this notation and when.</p> <p>(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!)</p> http://mathoverflow.net/questions/74004/what-does-the-in-algebra-stand-for/74013#74013 Answer by Georges Elencwajg for What does the σ in σ-algebra stand for? Georges Elencwajg 2011-08-29T23:42:54Z 2011-08-30T07:00:35Z <p>From <a href="http://books.google.com/books?id=WECVEDeljqgC&amp;printsec=frontcover&amp;dq=elstrodt+integrationstheorie&amp;hl=de#v=onepage&amp;q&amp;f=false" rel="nofollow">Elstrodt's book</a> <em>Maß- und Integrationstheorie</em>, pages 13-14:</p> <blockquote> <p>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 <a href="http://books.google.com/books?id=WECVEDeljqgC&amp;printsec=frontcover&amp;dq=elstrodt+integrationstheorie&amp;hl=de#v=onepage&amp;q&amp;f=false" rel="nofollow">1</a>, S. 5 und S. 23).<br> Eine entsprechende Terminologie ist üblich mit dem Vorsatz „$\delta$..." für abzählbare Durchschnitte (z.B.„$\delta$ -Ring"). </p> </blockquote> <p>My translation: </p> <blockquote> <p>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 <a href="http://books.google.com/books?id=WECVEDeljqgC&amp;printsec=frontcover&amp;dq=elstrodt+integrationstheorie&amp;hl=de#v=onepage&amp;q&amp;f=false" rel="nofollow">1</a>, p. 5 and p. 23).<br> A corresponding terminology is usual with the prefix „$\delta$-..." for denumerable intersections [Durchschnitte] (for example "$\delta$ -ring") </p> </blockquote> <p>(The reference is to Hausdorff's <em>Grundzüge der Mengenlehre.</em> published in 1914.) </p> <p><strong>To sum up:</strong> the excerpt says that $\sigma$ [=Greek <em>s</em>] and $\delta$[=Greek <em>d</em>] come from the German words <em>Summe</em> and <em>Durchschnitt</em>, whose English translations are respectively <em>sum</em> and <em>intersection</em>. </p>