Are the two definitions of the BGG category equivalent?

Question

Let $\mathfrak{g}$ be a finite dimensional complex semisimple lie algebra, the BGG category $\mathcal{O}$ is defined as the set of $\mathfrak{g}$-module $M$ such that

1. $M$ is finitely generated;
2. $M$ is a weight module;
3. $U(\mathfrak{n}^+)v$ lies in a finite dimensional subspace for every $v\in M$.

There is another definition: Let $D(\lambda)=\{\mu\in\mathfrak{h}^\ast,\mu\prec\lambda\}$ be a cone with vertex $\lambda$, the category $\mathcal{O}$ is defined consists of $\mathfrak{g}$-module $M$ with

1. $M$ is a weight module;
2. the weight of $M$ lies in finitely cones $D(\lambda_1)\cup\ldots\cup D(\lambda_k)$.

My question is : are the two definitions of the BGG category equivalent? I can show they share most common properties, only the "finitely generated" may not hold in the latter "definition".

Answer 1

I think your doubt is justified: Let $M$ be an infinite-dimensional vector space on which $\mathfrak{g}$ acts trivially. Then $M$ is a weight module and the unique weight (i.e. $0 \in \mathfrak{h}^*$) of $M$ lies in $D(0)$, but $M$ is not finitely generated.