ON THE UNIFORM KADEC-KLEE PROPERTY WITH RESPECT TO CONVERGENCE IN MEASURE

Let E(0, infinity) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace mu, and let E(M, mu) be the symmetric operator space associated with E (0, infinity). If E(0, infinity) has the uniform Kadec-Klee property w ith respect to convergence in measure then E(M, mu) also has this prop erty. In particular, if L(Phi) (0, infinity)(Lambda(phi)(0, infinity)) is a separable Orlicz (Lorentz) space then L(Phi)(M, mu)(Lambda(Phi)( M, mu)) has the uniform Kadec-Klee property with respect to convergenc e in measure. It is established also that E(0, infinity) has the unifo rm Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, infinity) satisfies G. Birkhoff's condition of uniform monotonicity.