  Given a von Neumann algebra $M$ equipped with a faithful normal strictly
semifinite weight $\varphi$, we develop a notion of Murray-von Neumann
dimension over $(M,\varphi)$ that is defined for modules over the basic
construction associated to the inclusion $M^\varphi \subset M$. For
$\varphi=\tau$ a faithful normal tracial state, this recovers the usual
Murray-von Neumann dimension for finite von Neumann algebras. If $M$ is either
a type $\mathrm{III}_\lambda$ factor with $0<\lambda <1$ or a full type
$\mathrm{III}_1$ factor with $\text{Sd}(M)\neq \mathbb{R}$, then amongst
extremal almost periodic weights the dimension function depends on $\varphi$
only up to scaling. As an application, we show that if an inclusion of diffuse
factors with separable preduals $N\subset M$ is with expectation $\mathcal{E}$
and admits a compatible extremal almost periodic state $\varphi$, then this
dimension quantity bounds the index $\text{Ind}{\mathcal{E}}$, and in fact
equals it when the modular operators $\Delta_\varphi$ and $\Delta_{\varphi|_N}$
have the same point spectrum. In the pursuit of this result, we also show such
inclusions always admit Pimsner-Popa orthogonal bases.
