Integrable kac moody module

Let $\mathfrak{g}(A)$ be the algebra of Kac moody algebra associated to a gcm, with its simple generators $e_i,f_i$ and Cartan $\mathfrak{h}$. A module $V$ is integrable if it is a weight module and every $e_i,f_i$ acts locally nilpotently. Explicitly,

$V={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{V}_{\mu },V_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\},$

and for every index $i$ and vector $v\in V$ there are positive integers $N_e,N_f$, depending on $i,v$, such that $e_i^{N_e}v=0$ and $f_i^{N_f}v=0$.

Only the weight-module convention from Kac moody category o is used. Finite-dimensional weight spaces, bounded-above support and finite generation are not part of integrability. Nor is a uniform nilpotence exponent required. The zero module is integrable, taking exponents one on its sole vector. Both raising and lowering conditions are required, independently; no AC is part of this definition.