In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G� ...
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 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
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
of the integral.
Axioms
We start by choosing a family
of bounded real functions (called ''elementary functions'') defined over some set
, that satisfies these two axioms:
*
is a linear space with the usual operations of addition and scalar multiplication.
* If a function
is in
, so is its
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
.
In addition, every function ''h'' in ''H'' is assigned a real number
, which is called the ''elementary integral'' of ''h'', satisfying these three axioms:
; Linearity
: If ''h'' and ''k'' are both in ''H'', and
and
are any two real numbers, then
.
; Nonnegativity
: If
for all
, then
.
; Continuity
: If
is a nonincreasing sequence (i.e.
) of functions in
that converges to 0 for all
in
, then
.or (more commonly)If
is an increasing sequence (i.e.
) of functions in
that converges to h for all
in
, then
.
That is, we define a continuous non-negative
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
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 function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having onl ...
s evidently satisfies the above axioms for elementary functions. Defining the elementary integral of the family of step functions as the (signed) 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 function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s as the elementary functions and the traditional
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G� ...
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
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, along with an appropriate function of
bounded variation, gives a definition of integral equivalent to the
Lebesgue–Stieltjes integral.
Sets of
measure zero
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
may be defined in terms of elementary functions as follows. A set
which is a subset of
is a set of measure zero if for any
, there exists a nondecreasing sequence of nonnegative elementary functions
in ''H'' such that
and
on
.
A set is called a set of
full measure if its complement, relative to
, 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
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.
Definition
Although the end result is the same, different authors construct the integral differently. A common approach is to start with defining a larger class of functions, based on our chosen elementary functions, the class
, which is the family of all functions that are the limit of a nondecreasing sequence
of elementary functions, such that the set of integrals
is bounded. The integral of a function
in
is defined as:
:
It can be shown that this definition of the integral is well-defined, i.e. it does not depend on the choice of sequence
.
However, the class
is in general not closed under subtraction and scalar multiplication by negative numbers; one needs to further extend it by defining a wider class of functions
with these properties.
Daniell's (1918) method, described in the book by Royden, amounts to defining the upper integral of a general function
by
:
where the infimum is taken over all
in
with
. The lower integral is defined in a similar fashion or shortly as
. Finally
consists of those functions whose upper and lower integrals are finite and coincide, and
:
An alternative route, based on a discovery by Frederic Riesz, is taken in the book by Shilov and Gurevich and in the article in Encyclopedia of Mathematics. Here
consists of those functions
that can be represented on a set of full measure (defined in the previous section) as the difference
, for some functions
and
in the class
. Then the integral of a function
can be defined as:
:
Again, it may be shown that this integral is well-defined, i.e. it does not depend on the decomposition of
into
and
. This turns out to be equivalent to the original Daniell integral.
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
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
Fatou's le ...
, and
Fubini's theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if th ...
may also readily be proved using this construction. Its properties are identical to the traditional Lebesgue integral.
Measurement
Because of the natural correspondence between sets and functions, it is also possible to use the Daniell integral to construct a
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
. If we take the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
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
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wi ...
.
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
. 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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
), one runs into practical difficulties using Lebesgue's construction that are alleviated with the Daniell approach.
The Polish mathematician
Jan Mikusinski has made an alternative and more natural formulation of Daniell integration by using the notion of absolutely convergent series. His formulation works for the
Bochner integral (the Lebesgue integral for mappings taking values in
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s). Mikusinski's lemma allows one to define the integral without mentioning
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
s. He also proved the change of variables theorem for multiple Bochner integrals and Fubini's theorem for Bochner integrals using Daniell integration. The book by Asplund and Bungart carries a lucid treatment of this approach for real valued functions. It also offers a proof of an abstract
Radon–Nikodym theorem using the
Daniell–Mikusinski approach.
See also
*
Lebesgue integral
*
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G� ...
*
Lebesgue–Stieltjes integration
References
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{{integral
Definitions of mathematical integration