In this paper we consider the modified maximal operator on the separable metric space. Define [Formula] and [Formula] respectively. We investigate in what parameter k the weak (1, 1)-inequality holds for Mk and Mk, uc in general metric space and Euclidean space. The proofs are sharper than the method of Vitali’s covering lemma. This attempt is partially done by Yutaka Terasawa [9] before. When we investigate Rd, we prove a new covering lemma of Rd. We also show that our condition on parameter k is sharp. In connection with this we consider the dual inequality of Stein type and its applications. © 2005 by the University of Notre Dame. All rights reserved.