When dealing with outer measure, I came up with this problem when extending a set function $\rho:\mathcal{C}\to [0,\infty]$ with $X=\bigcup\mathcal{C}$ and $\rho(\varnothing)=0$ to an outer measure by defining $$ \rho^{*}(E):=\inf\left\{ \sum_{i=1}^{\infty} \rho(C_i): C_i\in\mathcal{C} \text{ and } E\subset \cup_{i=1}^{\infty} C_i \right\}. $$
The family $\mathcal{C}$ is a subset of $2^X$ that contains $\varnothing$.
It seems fine but the cover for $X$ is an arbitrary cover. Is there any theorem to guarantee that we can always find a countable cover for any subset $E\in 2^X$?