Strict K-monotonicity and K-order continuity in symmetric spaces

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This paper is devoted to strict K- monotonicity and K-order continuity in symmetric spaces. Using the local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on [0,∞) implies that the embedding EL¹[0,∞) does not hold. We finish this article with a complete characterization of K-order continuity in a symmetric space E that is written using a notion of order continuity under some assumptions on the fundamental function of E.