In measure theory, what does the $\pi$ in $\pi$-system stand for? Also, what about the $\lambda$ in $\lambda$-system? I want to know why Dynkin chosen these names, and why these names make sense.
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5$\begingroup$ I assume the $\pi$ is for "product", since intersections/unions are typically seen as analogous to products/sums. (Likewise the $\sigma$ in $\sigma$-algebra is for "sum"). I don't know about $\lambda$. $\endgroup$– Nate EldredgeCommented Jun 7, 2017 at 20:19
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"D. P. Bertsekas and S. E. Shreve [39, p. 133] use the term Dynkin system, while P. Billingsley [43] and E. B. Dynkin [111] himself use the term $\lambda$-system. B. Fristedt and L. Gray [129, pp. 724–725] use the term Sierpinski class as they attribute Dynkin’s Lemma 4.11 below to W. Sierpinski [306], though they credit Dynkin with popularizing it. R. M. Blumenthal and R. K. Getoor [52] use the term d-system." [Charalambos&Kim]p135 footnote.
Dynkin did not mention why he used the term $\pi$-system in the book cited above(p.202 of [Dynkin2]) but mentioned the notion of $\pi$ and $\lambda$ systems as stated in a measure theoretic appendix. But as mentioned, another origin of the terms is "Sierpinski class"[Sierpinski]. Referring to this source, we can find that (p.14) he specifically discussed the motivation of this definition. To be more precise we not only need it to be "closed under multiplication" but also need the "multiplication be commutative". On the other hand if you examine the Kolmogorov consistency conditions in stochastic processes, you can actually figure out that the set class induced by the inverse image of a consistent finite dimensional distribution class is naturally a $\pi$-system but not necessarily a $\lambda$-system since they do not have to be closed under monotone limits.
On the other hand, independent $\pi$-systems $A_1,A_2\cdots $ generated independent $\sigma$-fields $\sigma(A_1),\sigma(A_2),\cdots$. When independent classes $A_1,A_2\cdots$ are not $\pi$-systems the corresponding generating $\sigma$-fields may fail be be independent. If we consider the generating $\sigma$-fields in terms of dual probability Hilbert spaces like [Small&McLeish] then the assumption that $A_1,A_2\cdots $ is $\pi$-system actually guarantee that the probability spans $$PS_i:=\mathrm{ProbSpan}\{\boldsymbol{1}_a,\forall a\in A_i\},i=1,2,\cdots$$ are mutually orthogonal as Hilbert spaces with compatible lattice structure, this means that the requirement of $\pi$-system contains some sort of minimality restriction.
So (from my point of view) the $\pi$-system is more or less a natural concept originated from the study of stochastic processes with orthogonal increments(in fact Dynkin proposed it under the title "Markov processes"). Yet the $\lambda$-system is more like a technical term filling the gap between $\pi$-system and the $\sigma$-field.
Wild guess below
Later I found a more "intuitive explanation". Since [Fristedt&Gray] p.725 made a historical remarks and said that the first occurrence is in [Dynkin] p.1. My curiosity drove me to look at the book. The book is a translated version, thus I took some time to figure out what was it like in the original Russian version(Дынкин, Евгений Борисович. "Основы теории марковских процессов." (1959).), Dynkin used these terms $\pi$ and $\lambda$ from nowhere, but a wild guess may be the capital form of Cyrillic alphabet $\pi\leftrightarrow П$ and $\lambda\leftrightarrow Л$ makes graphical variants of the product symbol $\prod$ but different enough. For the TRUE motivation, we have no way to know but ask Dynkin himself.
Reference
[Charalambos&Kim]Charalambos D. Aliprantis and Border, Kim C. Infinite Dimensional Analysis: A Hitchhiker's Guide. 3ed, Springer, 2006.
[Fristedt&Gray]Fristedt, Bert E., and Lawrence F. Gray. A modern approach to probability theory. Springer Science & Business Media, 2013.
[Dynkin1]Dynkin, Evgeniĭ Borisovich. Theory of Markov processes. Courier Corporation, 2012.
[Dynkin2]Dynkin, Evgenij Borisovic. Markov processes. Vol. 2. Springer,1965.
[Sierpinski]Un théorème général sur les familles d'ensembles. Autorzy. W. Sierpiński. Treść / Zawartość. http://matwbn.icm.edu.pl/ksiazki/fm/fm12/fm12117.pdf
[Small&McLeish]Small, Christopher G., and Don L. McLeish. Hilbert space methods in probability and statistical inference. Vol. 920. John Wiley & Sons, 2011.
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1$\begingroup$ this is interesting, but I wonder why symbols for mathematical objects need a reason to be chosen by the author; $\pi$, $\lambda$, $\rho$, $\sigma$, $\tau$ --- in papers I read these symbols tend to be just chosen because they are not yet taken, without a deeper reason. $\endgroup$ Commented Jun 19, 2017 at 13:20
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1$\begingroup$ @CarloBeenakker I agree that sometimes symbols are chosen without specific reasons, but when theories build up these meaningless symbols will usually be replaced by more carefully chosen new terms, "path-dependence of terms" is very rare in math, while this probably one... $\endgroup$– Henry.LCommented Jun 19, 2017 at 14:14
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$\begingroup$ Operations denoted by $\prod$, $\bigwedge$, $\bigcap$, etc usually share the same universal property (being some inverse limit of a discrete system) and so do their duals $\coprod$, $\bigvee$, $\bigcup$, (direct limits). So the choice $\pi$ to allude to $\cap$ is very natural. The reason for $\lambda$ is likely to be the one mentioned by Carlo Beenakker $\endgroup$ Commented Apr 13, 2020 at 14:36