[Math] Measurable functions on product space

measure-theoryproducts

Let $(\Omega, \mathcal{H}), (E, \mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces. Let $(E \times F, \mathcal{E} \otimes \mathcal{F})$ be a product space. Define the following three functions:

  1. $X:(\Omega,\mathcal{H}) \rightarrow (E,\mathcal{E})$
  2. $Y: (\Omega, \mathcal{H} \rightarrow (F,\mathcal{F})$
  3. $Z = (X,Y): (\Omega, \mathcal{H}) \rightarrow (E \times F, \mathcal{E} \otimes \mathcal{F})$

Now, I am trying to show that the following:
$X:(\Omega,\mathcal{H}) \rightarrow (E,\mathcal{E})$ and $Y:(\Omega, \mathcal{H}) \rightarrow (F,\mathcal{F})$ are measurable $\Leftrightarrow (X,Y): (\Omega,\mathcal{H}) \rightarrow (E \times F, \mathcal{E} \otimes \mathcal{F})$ is measurable.

Here is my current work:

$(\Rightarrow):$

Since X and Y are both measurable, we have
\begin{align*}
Z^{-1}(E \times F) &= \{w \in \Omega: Z(w) \in E \times F\}\\
&= \{w \in \Omega:(X(w),Y(w)) \in E \times F\}\\
&= \{w \in \Omega: X(w) \in E\text{ and }Y(w) \in F\}\\
&= \{w \in \Omega: X(w) \in E \} \cap \{w \in \Omega: Y(w) \in F\}\\
&= X^{-1}(E) \cap Y^{-1}(F),
\end{align*}
so the function $Z(w)$ is measurable for any $w \in \Omega$.

$(\Leftarrow):$ On this part I'm stuck so I just wrote out my assumptions and what I want to show:

We assume $Z(w)$ is measurable, so we have $Z^{-1}(A) \in \mathcal{H}$ for all $A \in \mathcal{E} \times \mathcal{F}.$ We need to show that X and Y are measurable, i.e. $X^{-1}(A) \in \Omega$ for all $B \in \mathcal{E}$ and $Y^{-1}(C) \in \Omega$ for all $C \in \mathcal{F}.$

How can I finish the rest of this proof? Unfortunately, I couldn't find anything like of this in my book (Probability and Stochastics by Cinlar)

Best Answer

Hint. Given $e \in \mathcal E$ (that is $e$ is a subset $e \subseteq E$, note, as Stefan commented, that you named the whole set $E$ already) we have $$ Z^{-1}(e \times F) = \{ \omega \in \Omega \mid Z(\omega) \in e \times F \} = \{\omega \in \Omega \mid X(\omega) \in e \} $$ (as $Y(\omega) \in F$ holds because of $Y \colon \Omega \to F$). Along the same lines $$ Z^{-1}(E \times f) = Y^{-1}(f), \quad f \subseteq F $$

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