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.
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 Bochner integral (Lebesgue integral for mappings taking values in Banach spaces). Mikusinski's lemma allows one to define integral without mentioning null sets. He also proved change of variables theorem for multiple integral for 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 Daniell–Mikusinski approach.
Read more about this topic: Daniell Integral
Famous quotes containing the words advantages over, advantages, traditional and/or formulation:
“A woman might claim to retain some of the child’s faculties, although very limited and defused, simply because she has not been encouraged to learn methods of thought and develop a disciplined mind. As long as education remains largely induction ignorance will retain these advantages over learning and it is time that women impudently put them to work.”
—Germaine Greer (b. 1939)
“But there are advantages to being elected President. The day after I was elected, I had my high school grades classified Top Secret.”
—Ronald Reagan (b. 1911)
“The community and family networks which helped sustain earlier generations have become scarcer for growing numbers of young parents. Those who lack links to these traditional sources of support are hard-pressed to find other resources, given the emphasis in our society on providing treatment services, rather than preventive services and support for health maintenance and well-being.”
—Bernice Weissbourd (20th century)
“You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.”
—Gerard Manley Hopkins (1844–1889)