We show exactly when the topology of convergence in measure in Banach ideal spaces is linear (equivalently, coarser than the norm topology). Next, we present the relationship between the Kadets–Klee and suitable monotonicity properties with respect to global convergence in measure. Applying these results, we characterize the Kadets–Klee property with respect to the global convergence in measure in infinite direct sums. We also prove the criteria of some related monotonicity properties in infinite direct sums. Furthermore, we solve the fundamental lifting (inheritance) problem completely for all these properties. We finish the paper with concrete examples showing how our general results can be applied.
Paweł Kolwicz (Sat,) studied this question.