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 $\mathbb{R}$:
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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 $I$.
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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!
Best Answer
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.