Let $\pi$ and $\mu$ be the target and proposal measures on $(X, \mathcal{X})$ respectively, with $\pi \ll \mu$. Suppose $\lambda$ is the reference measure on $(X, \mathcal{X})$ and that $\pi\ll \lambda$ and $\mu\ll\lambda$ with densities $$ \frac{d \pi}{d\lambda} = \frac{\tilde{p}_\pi}{\displaystyle \int \tilde{p}_\pi \, d\lambda} \qquad \qquad \frac{d \mu}{d \lambda} = \frac{\tilde{p}_\mu}{\displaystyle \int \tilde{p}_\mu d\lambda} $$ I would now like to find the usual expression for the Importance Sampling weights. For this, I consider the Radon-Nikodym derivative of $\pi$ with respect to $\mu$ $$ \begin{align} W = \frac{d \pi}{d \mu} &= \frac{d \pi}{d \lambda} \frac{d \lambda}{d \mu} && \text{Chain rule requires $\pi \ll\lambda \ll \mu$} \\ &= \frac{d \pi}{d\lambda} \left(\frac{d \mu}{d \lambda}\right)^{-1} && \text{This requires $\lambda \ll \mu$ and $\mu \ll \lambda$ i.e. equivalence} \\ &= \frac{\tilde{p}_\pi}{\displaystyle \int \tilde{p}_\pi \, d\lambda} \cdot \left(\frac{\tilde{p}_\mu}{\displaystyle \int \tilde{p}_\mu d\lambda}\right)^{-1} \\ &= \frac{\tilde{p}_\pi}{\tilde{p}_\mu} \cdot \left(\frac{\displaystyle \int \tilde{p}_\mu d \lambda}{\displaystyle \int \tilde{p}_\pi d\lambda}\right) \end{align} $$ Which makes sense because this agrees with the usual set-up. However, for this to work, we must have $\mu$ and $\lambda$ equivalent, but surely this is never the case in practice?
Is this derivation correct? I.e. do we need $\mu\ll\lambda$ and $\lambda \ll\mu$ to get our usual Importance Sampling weights?