Outer measures and weak type (1,1) estimates of Hardy-Littlewood maximal operators

We will introduce the $k$ times modified centered and uncentered Hardy-Littlewood maximal operators on nonhomogeneous spaces for $k>0$. We will prove the $k$ times modified centered Hardy-Littlewood maximal operator is weak type $(1,1)$ bounded with constant $1$ when $k \ge 2$ if the Radon measure of the space has ``continuitiy'' in some sense. In the proof, we will use the outer measure associated with the Radon measure. We will also prove other results of Hardy-Littlewood maximal operators on homogeneous spaces and on the real line by using outer measures