  Let $K$ denote the middle third Cantor set and ${\cal A}:= \{3^n : n = 0,1,2,
>... \} $. Given a real, positive function $\psi$ let $ W_{\cal A}(\psi)$
denote the set of real numbers $x$ in the unit interval for which there exist
infinitely many $(p,q) \in \Z \times {\cal A} $ such that $ |x - p/q| < \psi(q)
$. The analogue of the Hausdorff measure version of the Duffin-Schaeffer
conjecture is established for $ W_{\cal A}(\psi) \cap K $. One of the
consequences of this is that there exist very well approximable numbers, other
than Liouville numbers, in $K$ -- an assertion attributed to K. Mahler.
