Reason for product sigma algebra notation 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?

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.