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Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized as being inefficient or roundabout (see, e.g., the question “Why isn't integral defined as the area under the graph of function?”) and one might seek a way to define the Lebesgue integral directly, without mentioning measures at all (the Lebesgue measure can then retrospectively be defined as the integral of the characteristic function, making its properties obvious if those of the integral are correctly obtained).

Now some years ago, I taught a course on real analysis (which I hadn't myself written, conceived or organized) using the following definition of the Lebesgue integral on $\mathbb{R}$:

  • First, define a “step function” $\mathbb{R} \to \mathbb{R}$ as a finite linear combination of characteristic functions of intervals, and the integral of such a function as the linear form which takes the characteristic function of $I$ to the length of $I$.

  • Next, we say that $f\colon \mathbb{R} \to \mathbb{R}$ is integrable iff there exists a series $(\Sigma f_n)$ of step functions such that $\sum_{n=0}^{+\infty} \int|f_n| < +\infty$ and such that $f(x) = \sum_{n=0}^{+\infty} f_n(x)$ for every $x$ for which the RHS converges (edit: see below) absolutely; and when this is the case, we define $\int f := \sum_{n=0}^{+\infty}\int f_n$.

This provides a very short definition of what a Lebesgue-integrable function is, without going through the roundabout route of defining the measure first. Now of course it's not all rosy either: one has to check that this definition makes sense, and that it satisfies the usual properties of the Lebesgue integral. (And even if one knows in advance what the Lebesgue integral is, it's not quite obvious that this definition reconstructs it, because it's not trivial that one can construct a series $(\Sigma f_n)$ of step functions that converges to $f(x)$ at every point where it converges.)

(I also mentioned this definition in passing in the question “Can the Riemann integral be defined through a closure/completion process?”)

But anyway, my question is: who came up with this definition? Has anyone else seen it? What is its history? And are there any prominent courses in real integration that use it?

Edit / correction: Following Willie Wong's comment to Kostya_I's answer, I realize I had misremembered the definition, it's “$f(x) = \sum_{n=0}^{+\infty} f_n(x)$ for every $x$ for which the RHS converges absolutely” (i.e., $\sum_{n=0}^{+\infty} |f_n(x)| < +\infty$) rather than just “…for which the RHS converges”, so it appears that Jan Mikusiński is indeed the author of the definition I meant to write. But this raises the question of whether the definition I had actually written (with “converges” instead of “converges absolutely”) is different or whether this is irrelevant: if someone wants a crack at it, let them do so!

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    $\begingroup$ I think this approach is taken by Tom Apostol in his analysis book. I cannot remember who he credits it to (but hopefully there is credit if credit is due). $\endgroup$
    – Pedro
    Dec 1, 2022 at 11:22
  • $\begingroup$ @Pedro I just had a look. There is some similarity, but Apostol uses a.e. monotonic sequences of step functions rather than $L^1$-absolutely convergent series of step functions as in my question. Also, he defines “a measure zero” set beforehand (to be able to define a.e. monotonic convergence). I'm not sure how essential these differences are. $\endgroup$
    – Gro-Tsen
    Dec 1, 2022 at 12:19
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    $\begingroup$ Let me ask the obvious question. Who designed that course on real analysis that you taught? Have you asked him or her this question? $\endgroup$ Dec 1, 2022 at 13:19
  • $\begingroup$ Potentially relevant: mathoverflow.net/q/422774/3948 $\endgroup$ Dec 1, 2022 at 16:11
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    $\begingroup$ On absolute convergence issue: if you remove "absolute", proofs immediately start falling apart. For example, how do you check that a sum of two integrable functions is integrable? If it may happen that both corresponding series diverge at some point $x$, but once you combine them into one series, it converges. I don't know what is the class of functions you obtain this way, but it clearly seems a wrong thing to do... $\endgroup$
    – Kostya_I
    Dec 2, 2022 at 18:59

2 Answers 2

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This definition is due to Jan Mikusiński, see Mikusiński, Jan, The Bochner integral. Basel, Stuttgart: Birkhauser, 1978.

Mikusiński has co-authored another book on integration with Hartman in 1961, where a standard exposition of Lebesgue integration is given. So we may infer that Mikusiński's definition was invented between 1961 and 1978.

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    $\begingroup$ Note: Mikusinski's definition is different in a minor way from that listed in the OP. He only requires pointwise convergence where the series $\sum f_n(x)$ converges absolutely. $\endgroup$ Dec 1, 2022 at 15:51
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    $\begingroup$ Mikusinski also announced this characterization of Lebesgue integration by 1964 mathscinet.ams.org/mathscinet-getitem?mr=167585 (Sur une définition de l'intégrale de Lebesgue. (French) Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 203–204.) There is also a somewhat obscure reference one can find to "J. Mikusinski, An introduction to the theory of the Lebesgue and Bochner integrals, Univ. of Florida, Dept. of Math., Gainesville, 1964" in various publications. This seems to have been notes from a course/seminar he gave. $\endgroup$ Dec 1, 2022 at 16:08
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    $\begingroup$ (continuing from previous) see e.g. reference 13 in pnas.org/doi/abs/10.1073/pnas.63.2.266 (Brooks, PNAS 1969), which in context clearly refers to the the definition above. $\endgroup$ Dec 1, 2022 at 16:09
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    $\begingroup$ @WillieWong Ah, good point! I may have misremembered the definition I gave. Now I'm left to wonder if it really makes a difference. $\endgroup$
    – Gro-Tsen
    Dec 1, 2022 at 18:34
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    $\begingroup$ Let me add a comment about the logic of the Mikusinki approach. Apart from the simplification in the technical details (improved if one uses bricks instead of step functions), one gets the Lebesgue measurable sets for free, also the Bochner integral by slightly varying the concept of a brick. Of course, one can‘t avoid technicalities completely—they are bunched in proving that the integral is well-defined, i.e., independent of the choice of $(f_n)$. The OP might be interested in Joe Diestel‘s easily available review in the BAMS. $\endgroup$
    – klempner
    Dec 2, 2022 at 17:03
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This approach was used in the German Analysis (Calculus) textbook

MR0222221 Hans Grauert and Ingo Lieb, Differential- und Integralrechnung. Band I: Funktionen einer reellen Veränderlichen, Heidelberger Taschenbücher, Band 26 Springer-Verlag, Berlin-New York 1967.

and the review in Mathscinet says that their approach is original. (I was taught Analysis in the early 1970s from the Russian translation of this book. Another interesting feature is that they skip Riemann integral, and introduce this kind of Lebesgue integral from the very beginning.)

Remark. The MSN review of Mikusinski book (1978) mentioned in the answer by @Kostya_I (written by Halmos) credits him for this definition. This makes the question about the history of this definition more interesting.

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    $\begingroup$ Upon looking at Grauert-Lieb, though, their definition of Lebesgue integrable function (p.152) seems to be this: f is integrable if $\exists A$for any $\varepsilon>0$, there exist a lower semicontinuous $g$ and an upper semicontinuous $h$ with $h\leq f \leq g$ such that $|\int_I \psi - A|<\varepsilon$ for any piecewise constant $\psi$ with $h\leq\psi\leq g$. On surface, this looks rather different from Mikusiński's definition... $\endgroup$
    – Kostya_I
    Dec 1, 2022 at 16:11

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