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Question: In the definition of Product $\sigma$ algebra while considering the triple product is $\sigma_A ×\sigma_{(B×C)}$ a sigma algebra?

Meaning of Notation used

  • $\sigma_A$ is a sigma algebra on a set $A$, and
  • $B,C$ are two sets with the sigma algebras $\sigma_B$ and $\sigma_C$ respectively.
  • Now $\sigma_{(B×C)}$ is the product sigma algebra on $B×C$.

Now my question
Is $\sigma_A \times \sigma_{(B×C)}=\{(C,D):C\in \sigma_A \text{ and } D\in \sigma_{(B×C)}\}$ a sigma algebra on $A×B×C?$

My Guess : My guess was No/False, because when in books consider product sigma algebra on $A×B×C$ is generated by the above given collection (which I asked whether it is sigma algebra or not). Any help would be appreciated.

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  • $\begingroup$ If the cross products of two $\sigma$-algebras were a $\sigma$-algebra then the union of any two rectangles in $\mathbb R^2$ would have to be the cross product of two sets in $\mathbb R$. Is that possible ? Draw picture. $\endgroup$
    – Kurt G.
    May 28 at 9:44
  • $\begingroup$ Ordered pair $(C,D)$ with $C\subseteq A$ and $D\subseteq B\times C$ cannot be recognized as a subset of $A\times B\times C$ so cannot be an element of any $\sigma$-algebra on $A\times B\times C$. $\endgroup$
    – drhab
    May 28 at 9:50

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