# Sigma algebra generated by algebra

Let $$(X, \mathcal X , \mu)$$ and $$(Y, \mathcal Y, \nu)$$ are two meaure spaces and let $$\mathcal Z$$ is the sigma algebra that generated by $$P=AxB$$ sets where $$A \in \mathcal X$$ and $$B \in \mathcal Y$$. $$\mathcal {Z_0}$$ is family of finite unions of $$P$$ sets.

I’ve shown that $$\mathcal {Z_0}$$ is an algebra. I’d like to learn that is $$\mathcal Z$$ sigma algebra generated by $$\mathcal {Z_0}$$ algebra? How can I show it? I couldn’t understanf these concepts well. Thanks for any help.

## 1 Answer

Let $$\mathcal A$$ be the sigma algebra generated by $$\mathcal Z_0$$. Let us show that $$\mathcal A$$ is contained in $$\mathcal Z$$ and that $$\mathcal A$$ is contains in $$\mathcal Z$$. Since $$\mathcal Z$$ is a sigma algebra and it contains sets of the form $$A \times B$$ it also contains finite unions of these. Hence it contains $$\mathcal A$$ (because the latter is the smallest sigma algebra containing the sets $$A \times B$$).

On the other hand $$\mathcal A$$ is a sigma algebra which contains all finite unions of the sets $$A\times B$$ and hence it also contains sets of the type $$A\times B$$. Since $$\mathcal Z$$ is the smallest sigma algebra containing these sets we get $$\mathcal Z \subseteq\mathcal A$$