The Ordered Time Simplex: Borel Measurability and Volume

Let $k\ge1$ be a natural number and let $T>0$ be a real number. Write $\mathcal{B}_k$ and $\lambda_k$ for the $k$-fold Borel $\sigma$-algebra and product Lebesgue measure on $\mathbb{R}^k$, and $k!=1\cdot2\cdots k$ for the factorial. The ordered time simplex with horizon $T$ is the set $D_k(T)=\{(t_1,\dots,t_k)\in\mathbb{R}^k:\ 0<t_1<t_2<\dots<t_k\le T\}.$ Then $D_k(T)$ belongs to $\mathcal{B}_k$, and $\lambda_k\bigl(D_k(T)\bigr)=\frac{T^k}{k!}.$