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We study the structure of bounded simple weight $\mathfrak{sl}(\infty)$-,$\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, which have beenrecently classified in [6]. Given a splitting parabolic subalgebra$\mathfrak{p}$ of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$,$\mathfrak{sp}(\infty)$, we introduce the concepts of $\mathfrak{p}$-alignedand pseudo $\mathfrak{p}$-aligned $\mathfrak{sl}(\infty)$-,$\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, and give necessaryand sufficient conditions for bounded simple weight modules to be$\mathfrak{p}$-aligned or pseudo $\mathfrak{p}$-aligned. The existence ofpseudo $\mathfrak{p}$-aligned modules is a consequence of the fact that the Liealgebras considered have infinite rank.

On the structure of simple bounded weight modules of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$