Exercise: Generated Sigma-Algebra on a Four-Point Space

Prerequisites: Sigma-Algebras

Problem

Let $\Omega = \{1, 2, 3, 4\}$ and $\mathcal{C} = \{\{1, 2\},\, \{2, 3\}\}$.

1. Write out $\sigma(\mathcal{C})$ explicitly — list every set it contains.
2. How many elements does $\sigma(\mathcal{C})$ have? How does this compare to $|\mathcal{C}|$ and to the power set $2^\Omega$?
3. Is $\sigma(\mathcal{C})$ equal to the full power set $2^\Omega$? If not, identify an outcome in $\Omega$ that cannot be isolated (i.e. some outcome $\omega$ such that $\{\omega\} \notin \sigma(\mathcal{C})$).
4. In one sentence, describe the "information" represented by $\sigma(\mathcal{C})$ — what can an observer who knows only $\sigma(\mathcal{C})$ distinguish?

Hint

Start by taking all pairwise unions, intersections, and complements of the two generating sets. Then check whether the resulting collection is closed under the axioms. $|\sigma(\mathcal{C})|$ will always be a power of $2$ on a finite $\Omega$.