One of the main difficulties with the way the theory of the Lebesgue integral is normally developed 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 P.J. Daniell in the 1917 paper "A general form of integral" (Ann. of Math, 19, 279) that does not suffer from this deficiency, and has a few significant advantages as over the traditional formulation.
The basic idea involves the axiomatization of the integral. We start by choosing a family H of bounded real functions (called elementary functions) defined over some set X, that satisfies these two axioms:
- H is a linear space with the usual operations of addition and scalar multiplication.
- If a function h(x) is in H, so is its absolute value |h(x)|.
In addition, every function h in H is assigned a real number Ih, which is called the elementary integral of h, satisfying these three axioms:
- Linearity. If h and k are both in H, and α and β are any two real numbers, then I(αh + βk) = αIh + βIk.
- Nonnegativity. If h(x) ≥ 0, then Ih ≥ 0.
- Continuity. If hn(x) is a nonincreasing sequence (i.e. h1 ≥ ... ≥ hk ≥ ...) of functions in H that converges to 0 for all x in X, then Ihn -> 0.
That is, we define a continuous positive definite linear functional I over the space of elementary functions.
These elementary functions and their elementary integrals may be any set of functions and definitions of integrals that satisfy these axioms. The family of all step functions evidently satisfies the axioms for elementary functions. Defining the elementary integral of the family of step functions as the normal area underneath a step function evidently satisfies the axioms for an elementary integral. Applying the construction of the Daniell integral described further below using step functions as elementary functions produces a definition of an integral equivalent to the Lebesgue integral. Using the family of all continuous functions as the elementary functions and the traditional Riemann integral as the elementary integral is also possible, however, this will yield an integral that is also equivalent to Lebesgue's definition (however using these elementary functions simplifies the development of more advanced integrals such as the Stieltjes integral).
Sets of measure zero may be defined in terms of elementary functions as follows. A set Z which is a subset of X is a set of measure zero if for any ε > 0, there exists a nondecreasing sequence of nonnegative elementary functions hp(x) in H such that Ihp < ε and
sup h (x) ≥ 1
on Z. A set is called a set of full measure if its complement, relative to X, is a set of measure zero. We say that if some property holds at every point of a set of full measure (or equivalently everywhere except on a set of measure zero), it holds almost everywhere.
We can then proceed to define a larger class of functions, based on our chosen elementary functions, the class L+, which is the family of all functions that are the limit of a nondecreasing sequence hn of elementary functions almost everywhere, such that the set of integrals Ihn is bounded. The integral of a function f in L+ is defined as:
If = lim Ih
It can be shown that this definition of the integral is well-defined, i.e. it does not depend on the choice of sequence hn.
However, the class L+ is in general not closed under subtraction and scalar multiplication by negative numbers, but we can further extend it by defining a wider class of functions L such that every function φ(x) can be represented on a set of full measure as the difference φ = f - g, for some functions f and g in the class L+. Then the integral of a function φ(x) can be defined as:
∫ φ(x) dx = If - Ig
Again, it may be shown that this integral is well-defined, i.e. it does not depend on the decomposition of φ into f and g. This is the final construction of the Daniell integral.
Nearly all of the important theorems in the traditional theory of the Lebesgue integral, such as Lebesgue's theorem (on dominated convergence), the Riesz-Fischer theorem, Fatou's lemma, and Fubini's theorem may also readily be proved using this construction.
Because of the natural correspondence between sets and functions, it is also possible to use the Daniell integral to construct a theory of measure. If we take the characteristic function χ(x) of some set, then its integral may be taken as the measure of the set. This definition of measure based on the Daniell integral can be shown to be equivalent to the traditional Lebesgue measure.
This method of constructing the general integral has a few advantages over the traditional method of Lebesgue, particularly in the field of functional analysis. The Lebesgue and Daniell constructions are equivalent, as pointed out above, if ordinary finite-valued step functions are chosen as elementary functions. However, as one tries to extend the definition of the integral into more complex domains (e.g. attempting to define the integral of a linear functional), one runs into practical difficulties using Lebesgue's construction that are alleviated with the Daniell approach.
Source: G.E. Shilov and B.L. Gurevich, "Integral, Measure, and Derivative, a Unified Approach", trans. Richard A. Silverman.