Reflexivity of the Length Function

1. As in [l], 51 will denote a space of points P with a countably additive, non-negative measure y(E); all sets and functions considered will be 7-measurable; e will always denote a set of finite measure; E will be called purely infinite if y(E) = <» and eC-E implies y(e) =0. X is a length function if \(m) is defined, with O^X(m) fk °° , for every non-negative function u=u(P), and satisfies: (Ll) X(w) =0 if u(P) =0 for almost all P. (L2) \(u) 0. (L5) Ui(P) fku2(P) ^ ■ • • ior all P implies X(sup «„) = sup X(#„). uE will denote the restriction of u to E, i.e. uE(P) =u(P) if P is in E, =0 otherwise ;X(E) means X(m) with u(P) = 1 for Pin E, =0 otherwise; Un will denote min (u, N). X will be called continuous if for every