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