No. Suppose such $C_m$'s exist. Then without loss of generality $|C_m|\le1/3^m$ for all $m$. Let now $C:=\bigcap_{m=1}^\infty C_m^c$.
Then $|C|\ge1/2>0$, but $C_m\not\subseteq C$ for any $m$.
Iosif Pinelis
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