Questions about measurable mapping and product sigma algebra

Question

1. Suppose there are three measurable spaces $(\Omega, \mathbb{F})$, $(S_i, \mathbb{S}_i), i=1,2$, and two measurable mappings $f_i: \Omega \rightarrow S_i, i=1,2$. Is the mapping $f$ defined as $f(\omega):=(f_1(\omega), f_2(\omega))$ a measurable mapping from $(\Omega, \mathbb{F})$ to $(\prod_{i=1}^2 S_i, \prod_{i=1}^2 \mathbb{S}_i)$, where $\prod_{i=1}^2 \mathbb{S}_i$ is the product sigma algebra of $\mathbb{S}_i, i=1,2$?
2. Suppose there are four measurable spaces $(\Omega_i, \mathbb{F}_i), i=1,2$, $(S_i, \mathbb{S}_i), i=1,2$, and two measurable mappings $f_i: \Omega_i \rightarrow S_i, i=1,2$. Is the mapping $f$ defined as $f(\omega_1, \omega_2):=(f_1(\omega_1), f_2(\omega_2))$ a measurable mapping from $(\prod_{i=1}^2 \Omega_i, \prod_{i=1}^2 \mathbb{F}_i)$ to $(\prod_{i=1}^2 S_i, \prod_{i=1}^2 \mathbb{S}_i)$?
3. In Part 1 and Part 2, conversely, if $f$ is a measurable mapping, will $f_i, i=1,2$ be measurable mappings?
4. Can the statements in Part 1,2 and 3 be generalized to any collection of $(S_i, \mathbb{S}_i) i \in I$ and $(\Omega_i, \mathbb{F}_i) i \in I$?

Thanks and regards! Are there some websites or books that address these questions?

Answer 1

1. The product $\sigma$-algebra on $S_1\times S_2$ is generated by sets of the form $A\times B$, with $A\in \mathbb{S}_1$ and $B\in\mathbb{S}_2$. The function $f$ is measurable if and only if the inverse image of a measurable set is measurable, and it suffices to check that the inverse image of a measurable set in a specific generating set for the $\sigma$-algebra is measurable. So it suffices to see if $f^{-1}(A\times B)$, with $A$ and $B$ as above, is measurable.

   If $x\in f^{-1}(A\times B)$, then $f(x)\in A\times B$, so $f_1(x)\in A$ and $f_2(x)\in B$. Thus, $x\in f_1^{-1}(A)\cap f_2^{-1}(B)$. Conversely, if $x\in f_1^{-1}(A)\cap f_2^{-1}(B)$, then $f(x)\in A\times B$. So $f^{-1}(A\times B) = f_1^{-1}(A)\cap f_2^{-1}(B)$. Is this set in $\mathbb{F}$?

2. Same idea: what is $f^{-1}(A\times B)$? $(x,y)\in f^{-1}(A\times B)$ if and only if $x\in f_1^{-1}(A)$ and $y\in f_2^{-1}(B)$, so $f^{-1}(A\times B) = f_1^{-1}(A)\times f_2^{-1}(B)$. Is this set in the product $\sigma$-algebra $\mathbb{F}_1\times\mathbb{F}_2$?

3. Are the projection maps $\pi_i\colon (S_1\times S_2,\,\mathbb{S}_1\times\mathbb{S}_2)$ measurable? Is the composition of measurable functions measurable? What are $\pi_i\circ f$?

I'll leave 4 to you.