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In the German Wikipedia entry for $L^p$-Raum it is stated (Link)

Das $L$ in der Bezeichnung geht auf den französischen Mathematiker Henri Léon Lebesgue zurück, da diese Räume über das Lebesgue-Integral definiert werden.

"The $L$ in the name goes back to the French mathematician Henri Léon Lebesgue, since these spaces are defined by the Lebesgue integral."

However, in the English Wikipedia version,

They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910)

If we check the Riesz paper (1910) paper, this is how he introduces...

In der vorliegenden Arbeit wird die Voraussetzung der quadratischen Integrierbarkeit durch jene der Integrierbarkeit von $|f(x)|^p$ ersetzt; $p$ bedeutet eine beliebige, rationale oder irrationale Zahl $>1$.*) Jede Zahl $p$ bestimmt eine Funktionenklasse $\left[L^p\right]$. Die Rolle der Klasse $\left[L^2\right]$ übernehmen hier je zwei Klassen $\left[L^p\right]$ und $\left[\frac{p}{L^{p-1}}\right] ;$ sie haben die Eigenschaft, dass jede Funktion, die mit allen Funtetionen der einen Klasse integrierbare Produkte ergibt, sicher der andern Klasse angehört. Die Untersuchung dieser Funktionenklassen wird auf die wirklichen und scheinbaren Vorteile des Exponenten $p=2$ ein ganz besonderes Licht werfen; und man kann auch behaupten, dab sie für eine axiomatische Untersuchung der Funktionenräume brauchbares Material liefert.

In the present work, the condition of quadratic integrability is replaced by that of integrability of $|f(x)|^p$; $p$ means any rational or irrational number $>1$.*) Each number $p$ determines a class of functions $\left[L^p\right]$. The role of the class $\left[L^2\right]$ is played here by two classes each $\left[L^p\right]$ and $\left[\frac{p}{L^{p-1}}\right] ;$ they have the property that any function yielding integrable products with all functions of one class certainly belongs to the other class. The study of these classes of functions will throw a very special light on the real and apparent advantages of the exponent $p=2$; and one can also claim that it provides useful material for an axiomatic study of function spaces.

Translated with www.DeepL.com/Translator (free version)

The author does not associate the function class $L$ with the name Lebesgue. Is it possible that the explanation of the choice of $L$ is just like the $\operatorname{sinc}$ function which is popularly known as sinus cardinalis but without any solid source (Origin of $\operatorname{sinc}$ function)?
I mean the original author never explained his choice of $L$ but later people associated $L$ with Lebesgue.

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    $\begingroup$ Riesz certainly talks a lot about Lebesgue and how Lebesgue made the analysis possible. It looks at least plausible that this is where the name comes from. $\endgroup$
    – Michael Greinecker
    Commented Oct 24, 2022 at 18:28
  • $\begingroup$ Note that what is here written $$\left[\frac{p}{L^{p-1}}\right]$$ should actually be $$\left[L^{\frac{p}{p-1}}\right]$$. $\endgroup$
    – Gerald Edgar
    Commented Oct 25, 2022 at 7:47

1 Answer 1

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On page 453 of his 1910 paper Untersuchungen über Systeme integrierbarer Funktionen Riesz writes (in a footnote)

*) H. Lebesgue Sur les intégrales singulières (1909) appeared after the completion of this work. That paper touches on many of the points raised in sections 2 and 3 of the present paper, but only for the case $p=2$.

So it is not a stretch to assume that Riesz used $L^2$ in deference to Lebesgue, then generalized to $L^p$.

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    $\begingroup$ Also, the Jahrbuch review by Hellinger writes with emphasis: “This work is based on using, instead of the square integrable functions, the class $[L^p]\ (p>1)$ of all functions, whose $p$-th power is integrable in Lebesgue's sense.” Maybe this comes closest to saying that $L$ stands for Lebesgue. $\endgroup$
    – Francois Ziegler
    Commented Oct 24, 2022 at 21:22
  • $\begingroup$ Thanks. It is indeed very likely. The original author Riesz never explicitly stated what L stands for. It seems like the sinc story that we discussed a couple of years ago. $\endgroup$
    – ACR
    Commented Oct 25, 2022 at 0:28
  • $\begingroup$ @AChem E. Hilb-M.Riesz 1924 is an early article that explicitly ties the $L$ with Lebesgue, see p. 1191. You'll find more with a google books search for “Lebesgue class”. $\endgroup$
    – Francois Ziegler
    Commented Oct 28, 2022 at 4:25
  • $\begingroup$ @FrancoisZieglerm Thank you. $\endgroup$
    – ACR
    Commented Oct 28, 2022 at 4:41

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