Year: 1942
Type: article
Abstract: Introduction.S. Banach has introduced an integral which has no convergence properties, and wMch is defined for all bounded functions in (0, 1).Evidently this class does not contain the class (L) of Lebes- gue integrable functions.Since Banach integral is the integral without convergence properties, it will be desirable to define the Banach integral so that the class (B) of Banach integrable functions contains the class (L) and if Banach integral is pressed to have convergence properties of Lebesgue, then (B) reduces to (L).This is possible by the Jessen-Khintchine theorem.In the case of abstract-integral, it is desirable to define such integral.For this purpose, we have introduced the abstract Banach integral for which above relation holds for the abstract-Lebesgue integrals in the third and fourth papers.This is given in 1.In 2 we define the second Banach integral such that above relation holds for abstract Rie'mann integral, 3 contains a certain uniqueness theorem of above two integrals.4 contains that above consideration can be extended to the case where the value of the integral lies in a semi-vector lattice instead of real number field.
Institution: Tohoku University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W970891691