Showing that the union of two algebras will give us a disjoint finite union of sets.

Question

Suppose we have a space $X$ that could be anything and $\mathcal{A}_{1}$, $\mathcal{A}_{2}$ are algebras. How would I show that that algebra generated by $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ consists of just a finite unions that are disjoint and are of the form $B_{1} \cap B_{2}$ where $B_{1} \in \mathcal{A}_{1}$ and $B_{2} \in \mathcal{A}_{2}$. The help would be appreciated.

Answer 1

A strategy could be the following: denote by $\mathcal C$ the finite disjoint unions of sets of the form $B_1\cap B_2$ with $B_i\in\mathcal A_i$. We have to show that

• $\mathcal C$ is an algebra which contains both $\mathcal A_1$ and $\mathcal A_2$, and
• if $\mathcal A$ is an algebra containing $\mathcal A_1$ and $\mathcal A_2$, then it also contains $\mathcal C$.