Daniell integral

In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by Percy J. Daniell (1918) that does not suffer from this deficiency, and has a few significant advantages over the traditional formulation, especially as the integral is generalized into higher-dimensional spaces and further generalizations such as the Stieltjes integral. The basic idea involves the axiomatization of the integral.

We start by choosing a family ${\displaystyle H}$ of bounded real functions (called elementary functions) defined over some set ${\displaystyle X}$, that satisfies these two axioms:

In addition, every function h in H is assigned a real number ${\displaystyle Ih}$, which is called the elementary integral of h, satisfying these three axioms:

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