Hallo,
I have an enlarged sigma-field \sigma(\mathcal{F}\vee \sigma(X)) (that is the smallest sigma-field containing the sigma-fields \mathcal{F} \text{ and }\sigma(X)?
How does the generator of this (enlarged) sigma-field look like?
Is this sigma-field generated by sets of the form \{A \cap B: A \in\mathcal{F},B \in \sigma(X)\},
i.e.
\sigma(\{A \cap B: A \in\mathcal{F},B \in \sigma(X)\})=\sigma(\mathcal{F}\vee \sigma(X))?

Thanks in advance