University of Cambridge,
4 July 2014.
Abstract Lebesgue integration is a basic, essential component of analysis. Yet most definitions of Lebesgue integrability and integration are rather complicated, typically depending on a series of preliminary definitions. For instance, one of the most popular approaches involves the class of functions that can be expressed as an almost everywhere pointwise limit of an increasing sequence of step functions. Another approach constructs the space of Lebesgue-integrable functions as the completion of the normed vector space of continuous functions; but this depends on already having the definition of integration for continuous functions.
So we might wish for a short, direct description of Lebesgue integrability that reflects its fundamental nature. I will present two theorems achieving this.
The first characterizes the space L1[0, 1] by a simple universal property, entirely bypassing all the usual preliminary definitions. It tells us that once we accept two concepts — Banach space and the mean of two numbers — then the concept of Lebesgue integrability is inevitable. Moreover, this theorem not only characterizes the Lebesgue integrable functions on [0, 1]; it also characterizes Lebesgue integration of such functions.
The second theorem characterizes the functor L1 from measure spaces to Banach spaces, again by a simple universal property. Again, the theorem characterizes integration, as well as integrability, of functions on an arbitrary measure space.
Slides In this pdf file.
Related paper The categorical origins of Lebesgue integration (2020).