ON THE LIFTING PROPERTY. Il l BY A. IONESCU TULCEA AND C. IONESCU TULCEA

1. Let Z be a locally compact space and / i ^ O a positive Radon measure on Z. Let MR(Z} /X) be the Banach algebra of all bounded real-valued /x-measurable functions defined on Z, endowed with the norm/—»||/||oo = sup2 G^|/(0)| . Let CR(Z) be the subalgebraof MR{Z,\X) consisting of all bounded continuous functions on Z and 3C(Z) the subalgebra of all ƒ G CR (Z) having compact support. For two funct i o n s / a n d g defined on Z we shall write ƒ = g whenever ƒ and g coincide locally almost everywhere. Let now T:f—*T/ be a mapping of .MR (Z, /x) into M%(Z, fi). Properties of T such as those listed below will be considered in what follows: (I) 2 W ; (II) jfs=g implies Tf=T0; (HI) 2 W ; (IV) ƒ è O implies 7 } â 0 ; (V) Taf+PG = <*Tf+pTg; (VI) Tfg=TfTg; (VII) Tf=f if feC%(Z). A mapping T:f-*Tf of MR(Z, /X) into M%(Z, /x) satisfying (I)-(V) will be called a linear lifting of M£(Z, /x) ; if the condition (VI) is also verified the mapping will be called a lifting of MR (Z, /X). A strong linear lifting [strong lifting] of M%(Z, /x) is a linear lifting [lifting] which verifies also (VII). If T:f—>Tf is a lifting of M%(Z, JJ,) and A is a ju-measurable set then we shall denote by PT(A) the set defined by the equation T^^fpPrU)' For each z(~Z denote by VT(Z) the set of all parts PT(V) where V belongs to V(z) and is /x-measurable. If TT is the set of all parts pT{A)—N where A is /x-measurable and N is locally /xnegligible then TT is a topology on Z (this result is essentially due to J. Oxtoby and has been given by him in a lecture at Yale in the fall of 1960).