Suppose you are given a set $ \Omega $ and a collection $ \mathcal{G} $ of subsets of $ \Omega $. Assume further that $ A \subset \Omega $. Now let $ \sigma_{\Omega} (\mathcal{G}) $ denote the smallest sigma-algebra on $ \Omega $ containing $ \mathcal{G} $, and let $ \sigma_{A}(\mathcal{G} \ \cap A) $ denote the smallest sigma-algebra on A containing the collection $ \mathcal{G} \ \cap A $.
Is it true that $ \sigma_{A}(\mathcal{G} \ \cap A) = \sigma_{\Omega} (\mathcal{G}) \cap A $ ?
The inclusion " $ \subset $ " is clear, since if $ \mathcal{H} $ is a sigma-algebra on $ \Omega $ containing $ \mathcal{G} $, then $ \mathcal{H} \cap A $ is a sigma-algebra on $ A $ containing $ \sigma_{A}(\mathcal{G} \ \cap A) $. But what about the other inclusion?
Thanks for your help! Regards, Si