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,τ), the associated space E(M, τ) of τ-measurable operators has the same property.