I want to show that the order of integration is irrelevant as consequence of Fubini's theorem.Thus to say that for every given permutation the value of $\int_{\mathbb{R}}\dots \int_{\mathbb{R}} f(x_1,\dots, x_n) d\mu_1({\sigma(1)}) \dots d\mu_1(\sigma(n))$ remains the same (where $\sigma$ is a permutation of n elements and $\mu_1$ the one-deimensional Lebesgue-measure).
I suppose that since every permutation is writable as a product of transpositions, it is enough to show that for every transposition the value of the integral remains the same. What might be reduced to the fact that we show for two indices in direct neighborhood (i and i+1), that the former is the case. Do you think this is consistent? Or is there an easier way?
Greetings
Rico
Just apply Fubini's Theorem repeatedly to bring $d\mu_1(x_1)$ to the inner most integral; then apply Fubini's Theorem repeatedly to bring $d\mu_1(x_2)$ to the next integral, etc.