Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{ \text{ disc}}(\Gamma\backslash G)$ of $L^2(\Gamma\backslash G)$ with at most a finite multiplicity $m_{ \Gamma}(\pi)$. While $m_{ \Gamma}(\pi)$ is unknown in general, we are interested in its limit as $\Gamma$ is taken to be in a tower of lattices $\Gamma_1\supset \Gamma_2\supset\dots$. For a bounded measurable subset $X$ of the unitary dual $\widehat{ G}$, we let $m_{ \Gamma_n}(X)$ be the sum of the multiplicity $m_{ \Gamma_n}(\pi)$ of a representation $\pi$ over all $\pi$ in $X$. Let $H_X$ be the direct integral of the irreducible representations in $X$, which is also a module over the group von Neumann algebra $\mathcal{ L}\Gamma_n$. We prove: \begin{ center} $\lim\limits_{ n\to \infty}\cfrac{ m_{ \Gamma_n}(X)}{ \dim_{ \mathcal{ L}\Gamma_n}H_X}=1$, \end{ center} for any bounded subset $X$ of $\widehat{ G}$, when i) $\Gamma_n$'s are cocompact, or, ii) $G=\SL(n,\mathbb{ R})$ and $\{ \Gamma_n\}$ are principal congruence subgroups.
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