Suppose we have a function $f(x_1,...x_n)$ that is measurable and integrable under the Lebesgue measure. Now we want to look at the functions $F(x_i,x_j)= \int_{\mathbb{R}}\dots \int_{\mathbb{R}} f(x_1,\dots , x_n) d\mu_1 \dots d\mu_{i-1}$ and $G(x_i,x_j)= \int_{\mathbb{R}} \int_{\mathbb{R}} F(x_i,x_j) d\mu_i d\mu_j$. Where i and j are in direct neighborhood, thus to say that j equals i+1 and $\mu $ is the Lebesgue measure. Now my question is one that arose in the context of another question (Fubini and the order of integration). So I may cite what one of the participants said:
"What sort of measurability do you assume for $f(x_1,…,x_n)$, and does that imply that you have measurability as a function of two variables after you fix some variables, integrate other variables, leaving only two variables free?"
In first place I would've said that by the use of Tonelli's theorem the result is clear.
But now I'm not as sure as I was before, because by the use of Tonelli I only get statements about the inner integrals and not for the searched operations.
So how can I show, what the citation asks?
Greetings Rico