  Using the local approach to the global structure of a symmetric space $E$ we
establish a relationship between strict $K$- monotonicity, lower (resp. upper)
local uniform $K$-monotonicity, order continuity and the Kadec-Klee property
for global convergence in measure. We also answer the question under which
condition upper local uniform $K$-monotonicity concludes upper local uniform
monotonicity. Finally, we present a correlation between $K$-order continuity
and lower local uniform $K$-monotonicity in a symmetric space $E$ under some
additional assumptions on $E$.
