Magnetism of a misfit layered cobaltite [Ca$_2$Co$_{4/3}$Cu$_{2/3}$O$_4$]$_x^{rm RS}$[CoO$_2$] ($x sim$ 0.62, RS denotes a rocksalt-type block) was investigated by a positive muon spin rotation and relaxation ($mu^+$SR) experiment. A transition to an incommensurate ({sf IC}) spin density wave ({sf SDW}) state was found below 180 K (= $T_{rm C}^{rm on}$); and a clear oscillation due to a static internal magnetic field was observed below 140 K (= $T_{rm C}$). Furthermore, an anisotropic behavior of the zero-field $mu^+$SR experiment indicated that the {sf IC-SDW} propagates in the $a$-$b$ plane, with oscillating moments directed along the c axis. These results were quite similar to those for the related compound [Ca$_2$CoO$_3$]$_{0.62}^{rm RS}$[CoO$_2$], {sl i.e.}, Ca$_3$Co$_4$O$_9$. Since the {sf IC-SDW} field in [Ca$_2$Co$_{4/3}$Cu$_{2/3}$O$_4$]$_{0.62}^{rm RS}$[CoO$_2$] was approximately same to those in pure and doped [Ca$_2$CoO$_3$]$_{0.62}^{rm RS}$[CoO$_2$], it was concluded that the {sf IC-SDW} exist in the [CoO$_2$] planes.
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