Projection of sets in product $\sigma$-algebra I'd like some help in an exercise that asks to show by an example that given two $\sigma$-algebras, the projection of a set in the product $\sigma$-algebra need not to be in the corresponding $\sigma$-algebra.
Is there any simple example of that? I only managed to realize that if the set is countable, then the projection has to be in the corresponding $\sigma$-algebra.

consider the lebesgue $\sigma$-Algebra on $\mathbb{R},\mathcal{L}(\mathbb{R})$ and the product $\sigma$-Algebra $\mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R})$.
Now take a set $A \not\in \mathcal{L}(\mathbb{R})$ and consider $A \times \{0\}$.
Then $A \times \{0\} \in \mathcal{L}(\mathbb{R}) \otimes \mathcal{L}(\mathbb{R})$ but the projection is A
