  Let $(\cx,\,d,\,\mu)$ be a metric measure space and satisfy the so-called
upper doubling condition and the geometrically doubling condition. In this
paper, the authors show that for the maximal Calder\'on-Zygmund operator
associated with a singular integral whose kernel satisfies the standard size
condition and the H\"ormander condition, its $L^p(\mu)$ boundedness with
$p\in(1,\infty)$ is equivalent to its boundedness from $L^1(\mu)$ into
$L^{1,\infty}(\mu)$. Moreover, applying this, together with a new Cotlar type
inequality, the authors show that if the Calder\'on-Zygmund operator $T$ is
bounded on $L^2(\mu)$, then the corresponding maximal Calder\'on-Zygmund is
bounded on $L^p(\mu)$ for all $p\in(1,\infty)$, and bounded from $L^1(\mu)$
into $L^{1,\infty}(\mu)$. These results essentially improve the existing
results.
