Let $x := (x_1, \dots, x_n) \in \mathbb{R}^n$ be a vector of states with $n \in \mathbb{N}$ dimensions in the space of real numbers and let $f, g : \mathbb{R}^n \to [0, 1]$ be some differentiable functions.
Consider the expression: $$ L := \sum_{i=1}^n \int_{\mathbb{R}^n} g(x) \frac{\partial}{\partial x_i}f(x) dx. $$
Observe that one iteration of the summation in $L$ with $i = 1$ is: $$ \int_{\mathbb{R}^{n-1}} \left[ \int_{\mathbb{R}} g(x) \frac{\partial}{\partial x_1}f(x) dx_1 \right] d(x_2, \dots, x_n). $$
Could $L$ then be expressed as the following? $$ L = \sum_{i=1}^n \int_{\mathbb{R}^{n-1}} \left[\int_{\mathbb{R}} g(x)\frac{\partial}{\partial x_i}f(x)dx_i\right]\prod_{j \neq i}^n dx_j $$
Alternatively, if using an index set $I = \{1, \dots, n\} \subset \mathbb{N}$, would the following be better? $$ L = \sum_{i \in I} \int_{\mathbb{R}^{n-1}} \left[ \int_{\mathbb{R}} \frac{\partial}{\partial x_i}f(x) dx_i \right] \prod_{j \in I \setminus \{i\}} dx_j $$