Daniell integral

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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 in his 1918 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, 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.

1 The Daniell axioms
2 Definition of the Daniell integral
3 Properties
4 Measures from the Daniell integral
5 Advantages over the traditional formulation
6 See also
7 References

[edit] The Daniell axioms

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:

1. $H$ is a linear space with the usual operations of addition and scalar multiplication.

2. 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:

1. Linearity. If h and k are both in H, and $α$ and $β$ are any two real numbers, then $I (α h + β k) = α I h + β I k$ .

2. Nonnegativity. If $h(x) \ge 0$ , then $Ih \ge 0$ .

3. Continuity. If $h n (x)$ is a nonincreasing sequence (i.e. $h_1 \ge \ldots \ge h_k \ge \ldots$ ) of functions in $H$ that converges to 0 for all $x$ in $X$ , then $Ih_n \to 0$ .

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 over these functions which satisfy these axioms. The family of all step functions evidently satisfies the above 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 given 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. Doing the same, but using the Riemann-Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the Lebesgue-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 $h p (x)$ in H such that $I h p < ε$ and $\sup_p h_p(x) \ge 1$ 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.

[edit] Definition of the Daniell integral

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 $h n$ of elementary functions almost everywhere, such that the set of integrals $I h n$ is bounded. The integral of a function $f$ in $L +$ is defined as:

$If = \lim_{n \to \infty} Ih_n$

It can be shown that this definition of the integral is well-defined, i.e. it does not depend on the choice of sequence $h n$ .

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:

$\int_X \phi(x) dx = If - Ig\,$

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.

[edit] Properties

Nearly all of the important theorems in the traditional theory of the Lebesgue integral, such as Lebesgue's dominated convergence theorem, the Riesz-Fischer theorem, Fatou's lemma, and Fubini's theorem may also readily be proved using this construction. Its properties are identical to the traditional Lebesgue integral.

[edit] Measures from the Daniell integral

Because of the natural correspondence between sets and functions, it is also possible to use the Daniell integral to construct a measure theory. 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.

[edit] Advantages over the traditional formulation

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.

[edit] See also

[edit] References

Daniell, Percy John, 1918, "A general form of integral," Annals of Mathematics 19:: 279-94.
------, 1919, "Integrals in an infinite number of dimensions," Annals of Mathematics 20: 281-88.
------, 1919, "Functions of limited variation in an infinite number of dimensions," Annals of Mathematics 21: 30-38.
------, 1920, "Further properties of the general integral," Annals of Mathematics 21: 203-20.
------, 1921, "Integral products and probability," American Journal of Mathematics 43: 143-62.
Royden, H. L., 1988. Real Analysis, 3rd. ed. Prentice Hall.
Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8.

Integral
Riemann integral • Lebesgue integral • Daniell integral • Darboux integral • Henstock-Kurzweil integral • Haar integral • Lebesgue-Stieltjes integral • Riemann-Stieltjes integral