On the role of L-Baire functions in abstract measure and integration

Melvin J. Maron
1975 Glasgow Mathematical Journal  
Segal and Kunze [7], following Loomis [4], used this idea as the basis for a very efficient, elementary presentation of the theory of measure spaces (X, S, n) and their associated integral spaces (X, H?(S, n), \-dfi). Maron [5] then used it to study (not necessarily constructed) abstract integral spaces (X, £C, /) in the absence of any structure on X. In [7] , measurable functions and constructed integrals are presented in an "integral oriented" way to illuminate the role of L-Baire functions.
more » ... L-Baire functions. In this article we show how to use .L-Baire functions to give quick, informative proofs of the basic properties of measures and their associated integrals (on the class of summable, measurable functions) using the "measure oriented" definitions of Halmos [2] . The idea is to show that the objects defined in the "measure oriented" way coincide with an "integral oriented" counterpart, hence a fortiori have the desired properties (see (2.5), (2.6) and (5.5)). Having done this, it is easy to obtain a very sharp and general Riesz-Markov type theorem (6.3) which describes the 1-1 correspondence between the collection of all (not necessarily complete) a-finite measure spaces (X, S, n) and the collection of all integral spaces (X, SP, I) for which £C satisfies the hypothesis of Stone [8] : / e £C =>f A 1 e £C. To obtain this generality, we use the definition of an integral space (X, £?, I) given in [5] which avoids null sets by allowingi? «= [-oo, oo]*. A discussion of the results obtained and their proofs is given in § 7.
doi:10.1017/s0017089500002524 fatcat:xdlq45dc35cxrovfeofwkto3aa