Local approach to Kadec-Klee properties in symmetric function spaces

We prove several results concerning local approach to Kadec-Klee properties with respect to global (local) convergence in measure in symmetric Banach function spaces which may be of independent interest. Moreover, we prove characterizations of these properties in the Lorentz spaces. Finally, we show applications of Hg and Hl points to the local best dominated approximation problems in Banach lattices.