The question might be a little weird, and I don't know what I'm doing. I've been thinking about if an integral form of $$ L = \int_{A} f(x) dx + \int_{A^2} f(x)f(y) dxdy + \int_{A^3} f(x)f(y)f(z) dxdydz + \dots $$ Could be written concisely as one integral only. We can assume that $f : \mathbb{R} \to \mathbb{R}$ is Riemann integrable, it can be also assumed that $A \subset \mathbb{R}$, so taking the cartesian product will give us $\mathbb{R}^n$ vectors with some dimension $n$. The end goal is to be able to approximate such integral numerically. The numerical integration can be done by choosing random vectors of different lengths from some distribution and then evaluating the appropriate integrals from the sum with Monte Carlo integration.
The first solution I had in mind was, in practice the maximum length for a vector is finite and known (call it $n$), so we could make all vectors $\mathbb{R}^n$, where for lower dimension vectors, we substitute the missing dimensions with $0$. The problem is this would make the integration evaluate $0$ for such missing dimension, making the whole integral evaluate at $0$.
The second solution was to make a set to accommodate all the different-sized vectors and then integrate over the whole set somehow. Apparently Lebesgue integration is suppose to work well for "weird shaped" sets as anything is integrable if it is measurable? Let $\Omega_n$ be set of $n$-dimensional vectors and then let $$ \Omega = \cup_{n = 1}^{\infty} \Omega_n . $$ For the one-dimensional case, let $\mu$ be the usual area measure, and $\mu_n$ for $n > 1$ be a product of these one-dimensional area measures. Let the measure $\mu$ defined as $$ \mu(\mathbf{v}_n) = \sum_{n = 1}^{\infty} \mu_n(\mathbf{v}_n \cap \Omega), $$ where $\mathbf{v} \in \mathbb{R}^n$. Using these, we should be able to Lebesgue integrate $$ \int_{\Omega} G(\mathbf{v}_n) d \mu(\mathbf{v}_n), $$ where $G(\mathbf{v}_n)$ is expanded as $f(x)f(y)\dots f(z)$ for the respective dimension $n$.
So my question is such a Lebesgue integral possible? I barely know any measure theory or the details of Lebesgue integration. Any comment on this idea is highly appreciated.
Also, I would like to know if it is possible to Riemann integrate over the whole sum and not do it piece-wise, if not why?