Let $(X,S,\mu)$ be a measure space. we have
(1) $\mu$ is said to have the property $P_1$ if there is a countable family $\mathcal{C}$ of compact subsets of $X$ such that for every $U\in S$ and every $\epsilon >0$, there is $C\in\mathcal{C}$ such that $C\subseteq U$ and $\mu(U\setminus C)<\epsilon$.
(2) $\mu$ is said to have the property $P_2$ for every $U\in S$ and every $\epsilon >0$, there is a compact set $C\in X$ such that $C\subseteq U$ and $\mu(U\setminus C)<\epsilon$
I am getting confused whether $P_1\implies P_2$ or the converse ?
You have $P_1\implies P_2$, because $P_1$ guarantees that you can always find a compact set. The converse may fail, because in $P_1$ you want to restrict to a fixed countable family of compacts.