I was wondering why it is so common to denote the product sigma algebra with the same symbol that is used for tensor products. Is there a specific reason that this product symbol is used or was is just the lack of innovation?
2026-03-31 19:16:58.1774984618
Reason for product sigma algebra notation
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Let $S(\mathcal A)$ denote the simple functions on a sigma-algebra $\mathcal A$. Thus $\mathcal A$ can be identified with those functions in $S(\mathcal A)$ that take only the values $0$ or $1$.
Then $S(\mathcal A \otimes \mathcal B) = S(\mathcal A) \otimes S(\mathcal B)$, where the right hand $\otimes$ is the vector space tensor product.