The continuous map $Q_{t}\colon\sigma_{t}\to\sigma$ defined by

$$Q_{t}(w^{\prime},x^{\prime\prime})=\left(\left(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime}\right)w^{\prime},\lambda(t)x^{\prime\prime}\right)$$

restricts to a homeomorphism between the interior of $\sigma_{t}$ and the interior of $\sigma$. Further, its inverse maps the Lebesgue measure $b_{\sigma}^{-1}\lambda_{\sigma}$ on $\sigma$ to the measure

$$(Q_{t}^{-1})_{*}b_{\sigma}^{-1}\lambda_{\sigma}=\left(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime}\right)^{q}\lambda(t)^{p-q}b_{\sigma^{\prime}}^{-1}\lambda^{\prime}_{\sigma^{\prime}}\otimes|dx^{\prime\prime}|,$$

on $\sigma_{t}$, where $|dx^{\prime\prime}|$ is Lebesgue measure on ${\mathbb{R}}^{p-q}$ normalized by ${\mathbb{Z}}^{p-q}$.