Exercise: Sigma-Algebra Check

Prerequisites: Probability Space

Problem

Let $\Omega = \{a, b, c, d\}$ . For each of the following collections, determine whether it is a sigma-algebra on $\Omega$ . If it is not, state which axiom fails and give the offending set.

Collection A:

\mathcal{F}_A = \{\emptyset,\, \Omega,\, \{a, b\},\, \{c, d\}\}

Collection B:

\mathcal{F}_B = \{\emptyset,\, \Omega,\, \{a\},\, \{b, c, d\},\, \{a, b\},\, \{c, d\}\}

Collection C:

\mathcal{F}_C = \{\emptyset,\, \Omega,\, \{a\},\, \{b\},\, \{a, b\},\, \{c, d\},\, \{b, c, d\},\, \{a, c, d\}\}

Hint

For each collection, check the three axioms in order: (1) $\Omega \in \mathcal{F}$ , (2) closed under complementation, (3) closed under countable unions. A single failure disqualifies the collection.

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