This paper is a step by step account of the Riesz approach to the Lebesgue integral. Besides, motivating the use of ``almost everywhere'' tools, we eliminate unnecessary equivalences, and we give a simple representation of a complete space of integrable functions, usually missing from classical treatises.
"The Riesz Approach to the Lebesgue Integral and Complete Function Spaces." Real Anal. Exchange 27 (2) 635 - 660, 2001/2002.