A Generalization of the Bochner Integral by the Method of Ranked Spaces

Introduction A generalization of the integral for real valued functions by the method of ranked spaces was prqposed by Professgr Ki in 1956 (2 5) The generalizpd integra1 called the ER integral by him has been investigated by many researdhers (1 9L I10D In 1968 Professor S ishi dwe1ther the theoof the ER integral in a series of papers (4DWhere she showed cPletingthe original idea of Professor KUnugi that actually the space jof ER integrable functions is a completion in the sense of ranked spacesof the space pf si1e (or stq)) functions suitably defg preneighborhoods and ranks on Wbreover it was shom that the integra1(f)of an ER integrable function fL is given by the limit 1 m I(fi) ere fi is a Caudhy sequence in converging to f In this paper we follow the w)rk of ProfessoT Nakanishi replacing the reals by a Bch space X iewe study vtor valued integrals by the method of raled spaces The space of simple fmctions is defined as usua1 In order to define suitable preneighborhoods and ranl(s onwe separate the conditions of Professor Nakanishi into twoely convergences ae andl convergences with respect to a suitable quasi-senli b(ie a simm which satisfies a modified tria1e inequality as appeared in a paper by Amemiya-Airida 1D One advatage to do so is that our theory gives usual Bodhner integral andER type Bodhner integral by suita1)1y cbOosing the quasi-Si-no In a later paper (6 7) Professor Nakanishi fOrmuated a general theory of ccpletios of rahked spaces and showed again thatfbr real valued case the ranked space of simple functions has a special type (may be called a completion Z Whi(h has one-one correspondence with the space of ER integra1)1e fmctions Inour case if we choose for the quasi-si-mnn llll the id