Using the local approach to the global structure of a symmetric space , we establish a relationship between strict -monotonicity, lower (resp., upper) local uniform -monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform -monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between -order continuity and lower local uniform -monotonicity in a symmetric space under some additional assumptions on .
"Lower and upper local uniform -monotonicity in symmetric spaces." Banach J. Math. Anal. 12 (2) 314 - 330, April 2018. https://doi.org/10.1215/17358787-2017-0047