Exercise: Intersection of Sigma-Algebras

Prerequisites: Sigma-Algebras

Problem

Let $\mathcal{G}$ and $\mathcal{H}$ be two $\sigma$-algebras on the same sample space $\Omega$.

1. Prove that $\mathcal{G} \cap \mathcal{H}$ (the collection of subsets of $\Omega$ lying in both $\mathcal{G}$ and $\mathcal{H}$) is a $\sigma$-algebra.
2. Give a concrete example on $\Omega = \{1, 2, 3, 4\}$ where $\mathcal{G} \cup \mathcal{H}$ is not a $\sigma$-algebra. Identify the axiom that fails and exhibit a specific set missing from $\mathcal{G} \cup \mathcal{H}$.
3. In the example from part 2, compute $\sigma(\mathcal{G} \cup \mathcal{H})$ — the smallest $\sigma$-algebra containing both. Interpret this as combining two sources of information.

Hint

For part 1, verify the three axioms directly using the fact that $\mathcal{G}$ and $\mathcal{H}$ each satisfy them. For part 2, choose $\mathcal{G}$ and $\mathcal{H}$ that individually contain different "atoms" of $\Omega$. The failing axiom will be closure under unions.