The Lebesgue Integral over ℝⁿ of a Product of Functions of One Coordinate Each

In the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, assume the axiom of dependent choice. Let $n\in\mathbb{N}$ and let $h_{1},\dots,h_{n}\in\mathcal{M}^{+}(\mathbb{R},\mathcal{B}(\mathbb{R}))$, given as a function $j\mapsto h_{j}$ on $[n]$. Finite products in $[0,\infty]$ are the iterated operation of the product of Addition and Multiplication on [0,∞] (ii).

(i) Products of functions of one coordinate. The function $x\mapsto\prod_{j=1}^{n}h_{j}(x_{j})$ on $\mathbb{R}^{n}$ lies in $\mathcal{M}^{+}(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$, and in $[0,\infty]$

$$\int_{\mathbb{R}^{n}}\prod_{j=1}^{n}h_{j}(x_{j})\,d\lambda_{n}(x)=\prod_{j=1}^{n}\int_{\mathbb{R}}h_{j}\,d\lambda .$$

(ii) Restriction to Borel sets. Let $B_{1},\dots,B_{n}\in\mathcal{B}(\mathbb{R})$, given as a function $j\mapsto B_{j}$ on $[n]$. Then, in $[0,\infty]$,

$$\int_{\mathbb{R}^{n}}\prod_{j=1}^{n}(h_{j}\mathbf{1}_{B_{j}})(x_{j})\,d\lambda_{n}(x)=\prod_{j=1}^{n}\int_{B_{j}}h_{j}\,d\lambda .$$