Write $\{0,1\}^\mathbb{N}$ as cartesian product $\{0,1\}^E \times \{0,1\}^O$, where $E$ is the evens and $O$ is the odds. Your usual sigma-algebra is the product sigma-algebra that goes with this product.
So we need something like this: Write $\otimes$ for product sigma-algebra. Then is $$\mathcal{F}\otimes\left(\bigcap_{k=1}^\infty\mathcal{G}_k\right) = \bigcap_{k=1}^\infty\left(\mathcal{F}\otimes\mathcal{G}_k\right)$$
Write $\{0,1\}^\mathbb{N}$ as cartesian product $\{0,1\}^E \times \{0,1\}^O$, where $E$ is the evens and $O$ is the odds. Your usual sigma-algebra is the product sigma-algebra that goes with this product.