Let $ (\Omega, \mathcal{F}) $ be a measurable space. Let $ A, B \in \mathcal {F} $ with $ A \cap B = \emptyset $. Let $ \mathcal{A} \subset \mathcal{F} $ the smallest $ \sigma $-field containing $ A $ and not containing $ B $ and $ \mathcal {B} \subset \mathcal{F} $ the smallest $ \sigma $-field containing $ B $ and not containing $ A $.
Is it true that $ \mathcal{F} $ is the smallest sigma algebra containing $ \mathcal {A} \cup \mathcal {B}$? And if $|\Omega|=\infty$ ?
There is a counter example?
Thank´s.