Decomposition $J$-measurable

We say that a J-decomposition $\mathcal{A} = \{A_r\}_{r=1}^s$ of a J-measurable set $C$ refines a J-decomposition $\mathcal{C} = \{C_i\}_{i=1}^k$ when for every $r \in \{1, \cdots, s\}$, there exists $i \in \{1, \cdots, k\}$ such that $A_r \subseteq C_i$.

My question: Can each $C_i$ in the J-decomposition $\mathcal{C} = \{C_i\}_{i=1}^k$ of the J-measurable set $C$ be expressed as a finite union of elements $A_r$ from the J-decomposition $\mathcal{A} = \{A_r\}_{r=1}^s$ of $C$, where $\mathcal{A}$ refines $\mathcal{C}$?

My attempt: Let $i$ fixed, I tried to define $R=\{j: A_j \subset C_i \}$ and prove that $C_i=\bigcup_{j \in R} A_j$. The inclusion $\bigcup_{j \in R} A_j \subset C_i$ is immeadiate. I'm stuck in prove that the other inclusion also hold.