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Rethinking the Lebesgue integral. (English) Zbl 1229.26019

Summary: This article describes a new approach to the description of the space of Lebesgue integrable functions \(L^{1}(K)\), where \(K\) is a ball in \(\mathbb{R}^{n}\). The traditional approach enlarges the concept of integration and the class of integrable functions. As an afterthought (the Riesz-Fischer theorem), it is shown that the space of integrable functions is complete in the \(L^{1}\) norm. Today we know that the primary goal of the theory is to create a complete space, because most of the theorems of functional analysis require completeness. Accordingly we define \(L^{1}\) as the abstract completion in the \(L^{1}\) norm of the space \(C(K)\) of continuous functions. The elements of this completion are equivalence classes of Cauchy sequences of continuous functions; the remaining task is to assign to each element \(f\) of this abstract \(L^{1}\) a function \(f(x)\) defined almost everywhere, that is, except on a set that can be enclosed in an open set of arbitrary small volume. We say that \(f(x)\) represents \(f\) if there is a Cauchy sequence in \(f\) that converges to \(f(x)\) ae. We prove that every \(f\) is represented by some \(f(x)\), and that \(f(x)\) and \(g(x)\) are equal ae if and only if they represent the same element of \(L^{1}\). We then show how to derive the usual results of Lebesgue theory. In this approach measure is a derived concept; a set \(S\) is measurable if its characteristic function represents some element of \(L^{1}\). The usual properties of measurable sets follow. I hope to see this approach adopted in the teaching of the Lebesgue theory.

MSC:

26A42 Integrals of Riemann, Stieltjes and Lebesgue type
97I50 Integral calculus (educational aspects)
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A25 Integration with respect to measures and other set functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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