Solution: Sigma-Algebra Check

Exercise: Sigma-Algebra Check

Collection A

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

Axiom 1: $\Omega \in \mathcal{F}_A$ . ✓
Axiom 2: $\{a,b\}^c = \{c,d\} \in \mathcal{F}_A$ ; $\{c,d\}^c = \{a,b\} \in \mathcal{F}_A$ ; $\Omega^c = \emptyset \in \mathcal{F}_A$ . ✓
Axiom 3: $\{a,b\} \cup \{c,d\} = \Omega \in \mathcal{F}_A$ ; all other unions stay within $\mathcal{F}_A$ . ✓

$\mathcal{F}_A$ is a sigma-algebra. It represents the coarse information "which pair did the outcome land in?" — identical in structure to the

\mathcal{G}

example in the parent lesson.

Collection B

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

Axiom 1: $\Omega \in \mathcal{F}_B$ . ✓
Axiom 2: $\{a\}^c = \{b,c,d\} \in \mathcal{F}_B$ ; $\{a,b\}^c = \{c,d\} \in \mathcal{F}_B$ . ✓
Axiom 3: $\{a\} \cup \{c,d\} = \{a,c,d\}$ . Is $\{a,c,d\} \in \mathcal{F}_B$ ? No. ✗

$\mathcal{F}_B$ is not a sigma-algebra. Axiom 3 fails: the union

\{a\} \cup \{c,d\} = \{a,c,d\}

is not in

\mathcal{F}_B

Note also that axiom 2 would fail on $\{a,c,d\}$ if it were added, since its complement $\{b\}$ is also missing. Fixing a broken sigma-algebra usually requires adding several sets at once.

Collection C

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

Axiom 1: $\Omega \in \mathcal{F}_C$ . ✓
Axiom 2: $\{a\}^c = \{b,c,d\} \in \mathcal{F}_C$ ; $\{b\}^c = \{a,c,d\} \in \mathcal{F}_C$ ; $\{a,b\}^c = \{c,d\} \in \mathcal{F}_C$ . ✓
Axiom 3: $\{a\} \cup \{b\} = \{a,b\} \in \mathcal{F}_C$ ; $\{a\} \cup \{c,d\} = \{a,c,d\} \in \mathcal{F}_C$ ; $\{b\} \cup \{c,d\} = \{b,c,d\} \in \mathcal{F}_C$ . All other unions are $\Omega$ or already-listed sets. ✓

$\mathcal{F}_C$ is a sigma-algebra. Notice it contains 8 sets but

|\Omega| = 4

, so the power set has

2^4 = 16

elements —

\mathcal{F}_C

is a strict sub-sigma-algebra of the full power set. It represents the information "is

a

the outcome?", "is

b

the outcome?", and their combinations, but cannot distinguish

c

from

d

Takeaways

To verify a sigma-algebra, check all three axioms systematically. A single failure — even one missing complement — disqualifies the collection.
The smallest sigma-algebra on $\Omega$ is $\{\emptyset, \Omega\}$ ; the largest is the power set $2^\Omega$ . Everything in between represents some coarsening of information.
When a collection fails axiom 3, the fix is not just to add the missing union — you typically also need its complement (axiom 2), which may generate further required unions. The collection "generated" by a family of sets is the smallest sigma-algebra containing all of them.