Let be a -finite measure space. The space of -measurable functions on is denoted by , and if , then denotes , where is the equivalence class of functions which differ from on sets of measure 0, and for , is the -norm.
In the paper, denotes a complete -finite sub-algebra of , and the operator is said to be a conditional expectation operator if for , , is -measurable and , whenever the integrals are finite. If , then the set of conditional multipliers is defined to be , denotes the multiplication operator , and if is a non-singular measurable transformation with measure which is absolutely continuous with respect to , the derivative is denoted by . If denotes the composition operator, , then is defined to be , and
, where . The results of this paper include:
is a Banach space, where , if and , and , if , , and is a countable set of pairwise disjoint -atoms such that ;
characterizations indicating that if and only if , where ; if and only a.e on and , where .