Let $f:\mathbb{R}^n\to\mathbb{R}$ be such that $f(x_1,\dots,x_n)=f(x_{\sigma(1)},\dots,x_{\sigma(n)})$ for every $n$-permutation $\sigma$, and suppose that $I=\int_{\mathbb{R}^n}f(x_1,\dots,x_n)\,dx_1\dots dx_n$ is finite.
Conjecture. There is a partition of $\mathbb{R}^n$ into $n!$ disjoint sets $S_1,\dots,S_{n!}$ such that $$ I=n!\int_{S_1}f(x_1,\dots,x_n)\,dx_1\dots dx_n \, . $$ I would like to receive feedback on this conjecture (proof or counterexample) as well as a description of the sets $S_i$.