Lifting of Kadec-Klee properties to symmetric spaces of measurable operators

We show that if E is a separable symmetric Banach function space on the positive half-line, then E has the Kadec-Klee property (respectively, uniform Kadec-Klee property) for local convergence in measure if and only if, for every semifinite von Neumann algebra (M, 'r), the associated space E(M, 'r) of 'r-measurable operators has the same property.