Let
$\left( \text{ }\!\!\chi\!\!\text{ ,}\,d,\,\mu \right)$
be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that
$\mu \left( \left\{ x \right\} \right)\,=\,0$
for all
$x\,\in \,\text{ }\!\!\chi\!\!\text{ }$
. In this paper, we show that the boundedness of a Calderón–Zygmund operator
$T$
on
${{L}^{2}}\left( \mu \right)$
is equivalent to that of
$T$
on
${{L}^{p}}\left( \mu \right)$
for some
$p\,\in \,\left( 1,\,\infty \right)$
, and that of
$T$
from
${{L}^{1}}\left( \mu \right)$
to
${{L}^{1,\,\infty }}\left( \mu \right)$
. As an application, we prove that if
$T$
is a Calderón–Zygmund operator bounded on
${{L}^{2}}\left( \mu \right)$
, then its maximal operator is bounded on
${{L}^{p}}\left( \mu \right)$
for all
$p\,\in \,\left( 1,\,\infty \right)$
and from the space of all complex-valued Borel measures on
$\text{ }\!\!\chi\!\!\text{ }$
to
${{L}^{1,\,\infty }}\left( \mu \right)$
. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.