Extension of a theorem of Bochner on expressing functionals as Riemann integrals

Brockway McMillan, Paco Lagerstrom
1945 Bulletin of the American Mathematical Society  
Introduction. S. Bochner 1 has shown that an additive homogeneous functional defined over a sufficiently large class C of functions can be realized as a Riemann integral with respect to a finitely additive measure V in the space X over which the functions are defined. His proof makes use of the fact that the constant function belongs to C, as a result, V(X) is finite. It is the purpose of this note to show that a similar theorem holds even when V(X) turns out to be infinite. A modification of
more » ... A modification of Bochner's proof would suffice for this stronger theorem. We have chosen rather to treat it as a problem of extending the domain of definition of the given functional. Throughout we have used the symbol -> to be read as "implies." The equality ss is used to denote an equality which holds by definition. Notations. We consider a space X of points x, and real-valued point functions/, g, • • • over X. Given/, g, and real numbers a, 6, we shall write
doi:10.1090/s0002-9904-1945-08323-2 fatcat:e27nclgtq5fkdp2j2kvxbcvqsm