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

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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