Lebesgue Measure on a Concatenated Euclidean Space: Iterated Integration over the Two Factors

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $q,p\in\mathbb{N}$, let $\iota^{q,p}:\mathbb{R}^{q}\times\mathbb{R}^{p}\to\mathbb{R}^{q+p}$ be the concatenation, and let $\lambda_{q},\lambda_{p},\lambda_{q+p}$ be Lebesgue measure on the Borel sets of $\mathbb{R}^{q}$, $\mathbb{R}^{p}$ and $\mathbb{R}^{q+p}$.

(Iterated integration)¶ Let $f:\mathbb{R}^{q+p}\to[0,\infty]$ be Borel. For every $u\in\mathbb{R}^{q}$ the function $v\mapsto f(\iota^{q,p}(u,v))$ on $\mathbb{R}^{p}$ is Borel, the function $u\mapsto\int_{\mathbb{R}^{p}}f(\iota^{q,p}(u,v))\,\lambda_{p}(dv)$ on $\mathbb{R}^{q}$ is Borel, and