Given a Young function $\Phi$, we study the existence of copies of $c_0$ and $\ell _{\infty }$ in $\mathop {\mathrm cabv}\nolimits _{\Phi} (\mu ,X)$ and in $\mathop {\mathrm cabsv}\nolimits _{\Phi } (\mu ,X)$, the countably additive, $\mu $-continuous, and $X$-valued measure spaces of bounded $\Phi $-variation and bounded
$\Phi$-semivariation, respectively.