Question on Sigma Algebras and Non-Finite Sets

[BrainHurts]

Homework Statement

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

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?

 

[Dick]

Homework Statement

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

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?

No, no need for infinite sets. Can you show the union of those two is not a sigma algebra?

 

[BrainHurts]

OK I'm going to do all the four steps

[itex]\mathcal{A}_1 \cup \mathcal{A}_2 = \{ \emptyset, \{ 1,2,3 \}, \{ 1 \}, \{ 2,3 \}, \{2 \} , \{ 1,3\} \}[/itex]

1) it is clear that [itex] \emptyset, \{1,2,3\} [/itex]are in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

2) so if [itex]A \in \mathcal{A}_1 \cup \mathcal{A}_2 [/itex], then [itex]A^c \in \mathcal{A}_1 \cup \mathcal{A}_2 [/itex].

I think this is satisfied, e.g. if [itex] A = \{ 1 \}[/itex] , then [itex]A^c = \{ 2,3 \} [/itex] and both are in [itex]\mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

3) if [itex] B_1, ... B_n \in \mathcal{A}_1 \cup \mathcal{A}_2 [/itex] then both

[itex] \bigcup_{i=1}^n A_i [/itex] and [itex] \bigcap_{i=1}^n A_i [/itex] are both in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

I think this is it! I just came up with it now,

so if I take [itex]A_1 = \{ 2,3 \} [/itex] and [itex] A_2 \{1,3\} [/itex] then the intersection is [itex] \{ 3 \} [/itex] and that's not in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]. Is this right? so it fails the condition that [itex] \bigcup B_i[/itex] is not in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

 

[BrainHurts]

Sorry I meant [itex] \bigcap B_i [/itex] is not in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

 

[Dick]

Sorry I meant [itex] \bigcap B_i [/itex] is not in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex]

Sure. You can get {3} by intersections or unions and complements of sets in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex] but it's not in [itex] \mathcal{A}_1 \cup \mathcal{A}_2 [/itex].