1. The problem statement, all variables and given/known data
Find a set X such that [itex] \mathcal{A}_1 \text{ and } \mathcal{A}_2 [/itex] are [itex]\sigma[/itex]-algebras where both [itex] \mathcal{A}_1 \text{ and } \mathcal{A}_2 [/itex] consists of subsets of X. We want to show that there exists such a collection such that [itex]\mathcal{A}_1 \cup \mathcal{A}_2[/itex] is not a [itex]\sigma[/itex] - algebra
3. The attempt at a solution
So here's what I'm thinking. I feel like for sure we need to fail the condition of Countable additivity.
I'm using a simple example like [itex]X = \{1,2,3\}[/itex] and I chose something [itex]\mathcal{A}_1 = \left\{\emptyset,\{1,2,3\}, \{1\}, \{2,3\} \right\} [/itex]
and [itex]\mathcal{A}_2 = \left\{\emptyset,\{1,2,3\}, \{2\}, \{1,3\} \right\} [/itex]
I have shown that both [itex] \mathcal{A}_1 [/itex] and [itex] \mathcal{A}_2 [/itex] are [itex] \sigma [/itex] algebras.
Am I on the right track here? Should I think of non-finite sets?
Find a set X such that [itex] \mathcal{A}_1 \text{ and } \mathcal{A}_2 [/itex] are [itex]\sigma[/itex]-algebras where both [itex] \mathcal{A}_1 \text{ and } \mathcal{A}_2 [/itex] consists of subsets of X. We want to show that there exists such a collection such that [itex]\mathcal{A}_1 \cup \mathcal{A}_2[/itex] is not a [itex]\sigma[/itex] - algebra
3. The attempt at a solution
So here's what I'm thinking. I feel like for sure we need to fail the condition of Countable additivity.
I'm using a simple example like [itex]X = \{1,2,3\}[/itex] and I chose something [itex]\mathcal{A}_1 = \left\{\emptyset,\{1,2,3\}, \{1\}, \{2,3\} \right\} [/itex]
and [itex]\mathcal{A}_2 = \left\{\emptyset,\{1,2,3\}, \{2\}, \{1,3\} \right\} [/itex]
I have shown that both [itex] \mathcal{A}_1 [/itex] and [itex] \mathcal{A}_2 [/itex] are [itex] \sigma [/itex] algebras.
Am I on the right track here? Should I think of non-finite sets?
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