By setting $w_{0}=b_{0}^{-1}(1-\sum_{i=1}^{p}b_{i}w_{i})$, we can identify $\sigma$ with the simplex $\sum_{1}^{p}b_{i}w_{i}\leq 1$ in ${\mathbb{R}}_{+}^{p}$. The normalized Lebesgue measure on $\sigma$ is then given by $\lambda_{\sigma}=b_{\sigma}^{-1}|dw_{1}\wedge\dots\wedge dw_{p}|$.