  We consider the BGG category $\mathcal{O}$ of a quantized universal
enveloping algebra $U_q(\mathfrak{g})$. We call a module $M\in \mathcal{O}$
tensor-closed if $M\otimes N\in\mathcal{O}$ for any $N\in \mathcal{O}$. In this
paper we prove that $M\in \mathcal{O}$ is tensor-closed if and only if $M$ is
finite dimensional. The method used in this paper applies to the unquantized
case as well.
