The Lebesgue Integral as an Improper Riemann Integral

Abstract

1. The purpose of this note is to point out that the Lebesgue integral of a real function, summable over a measurable set in some real Euclidean n-space, can be defined as a (one-dimensional) improper Riemann integral. This is in line (though in a different direction) with recent literature exhibiting the Lebesgue integral as a special case of the generalized Riemann integral (d., e.g., [1, p.289]). A treatment of improper Riemann integral can be found, e.g., in [41, Chapters 28, 29. 2. Theorem. Let f be a real function, defined and measurable on a measurable set S in some real Euclidean n-space. For every real y, let