Coordinate maps and permutations are measurable on finite products

By induction on $n$, the finite product sigma-algebra is generated by the $n$-coordinate rectangles. The base case is $n=1$. At the induction step, let $\mathcal{S}$ be the sigma-algebra generated by $(n+1)$-coordinate rectangles. For each measurable last-coordinate set $B$, the class of first-coordinate sets $A$ with $A\times B\in \mathcal{S}$ is a sigma-algebra: it contains the whole first product since the whole-first-product times $B$ is a coordinate rectangle, and complements and countable unions follow by relative difference inside that set and by countable unions. It contains the $n$-coordinate rectangles by construction, so it contains the recursively defined $n$-fold product sigma-algebra. Thus $\mathcal{S}$ contains every rectangle used in the recursive binary product. Conversely, each $(n+1)$-coordinate rectangle belongs to that recursive product sigma-algebra by its defining binary rectangle clause, so the two generated sigma-algebras agree.

For a generating rectangle $E_1\times \cdots \times E_n$, its preimage under $f$ is ${\bigcap }_{i=1}^n{f}_i^{-1}(E_i)$. Each set in this finite intersection belongs to $\mathcal{F}$ by [A1], so the preimage belongs to $\mathcal{F}$ by [F4]. The class of target sets whose preimages under $f$ lie in $\mathcal{F}$ is a sigma-algebra because preimages preserve complements and countable unions; by step 1.1 it contains the generators, so it contains the product sigma-algebra and [F3] makes $f$ measurable.

For $\pi$ a permutation, ${\tau }_{\pi }^{-1}(E_1\times \cdots \times E_n)=E_{{\pi }^{-1}(1)}\times \cdots \times E_{{\pi }^{-1}(n)}$, which is a generating rectangle by [F2]. The same generated-sigma-algebra argument of step 2.1 proves that ${\tau }_{\pi }$ is measurable.