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.
, that satisfies these two axioms: In addition, every function h in H is assigned a real number
, which is called the elementary integral of h, satisfying these three axioms: That is, we define a continuous non-negative linear functional
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.
Sets of measure zero may be defined in terms of elementary functions as follows.
, there exists a nondecreasing sequence of nonnegative elementary functions
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.
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
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
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
Daniell's (1918) method, described in the book by Royden, amounts to defining the upper integral of a general function
by The lower integral is defined in a similar fashion or, in short, as
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.
that can be represented on a set of full measure (defined in the previous section) as the difference
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
This turns out to be equivalent to the original Daniell integral.
Its properties are identical to the traditional Lebesgue integral.
Because of the natural correspondence between sets and functions, it is also possible to use the Daniell integral to construct a measure theory.
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.
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 spaces)[citation needed].
Mikusinski's lemma allows one to define the integral without mentioning null sets.
The book by Asplund and Bungart carries a lucid treatment of this approach for real valued functions.
It also offers a proof of the abstract Radon–Nikodym theorem using the Daniell–Mikusinski approach.