What does $\bigotimes$ mean in Sigma-algebras?

Question

What does $\otimes$ mean in $\sigma$-algebras?

Such as:

$$B([0,t]) \otimes F_t$$

where $F_t$ is part of filtration.

Here it's denoted $\times$:

https://mathoverflow.net/questions/176622/progressively-measurable-vs-adapted

So is it a Cartesian product?

Answer 1

Usually $\otimes$ stands for the product of $\sigma$--algebra, that is. if $\mathcal{A}$ and $\mathcal{B}$ are $\sigma$--algebras, $\mathcal{A}\otimes\mathcal{B}$ is the minimal $\sigma$--algebra that contains the sets $A\times B$, where $A\in\mathcal{A}$ and $B\in\mathcal{B}$.

In your case, $B([0,t])$ is the Borel $\sigma$--algebra in $[0,t]$ and $\mathcal{F}_t$ is the $\sigma$--algebra generated by $\{X_s:0\leq s\leq t\}$.

Answer 2

A kind of Cartesian product, yes: $\mathcal A\otimes \mathcal B$ is the $\sigma$-algebra generated by the sets $A\times B$ for $A\in\mathcal A, \, B\in\mathcal B$.