Question

Let F and G be two sigma-fields on Ω. Prove that F ∩ G is also a sigma-field on Ω.

Show by example that F ∪ G may fail to be sigma-field if Ω = {1, 2, 3}.

Answer 1

$\emptyset \in F,\emptyset \in G\Rightarrow \emptyset \in F\cap G\\A\in F\cap G\Rightarrow A\in F,A\in G\Rightarrow \bar{A}\in F,\bar{A}\in G\Rightarrow \bar{A}\in F\cap G\\A_i\in F\cap G\Rightarrow A_i\in F,A_i\in G\Rightarrow \bigcup{A_i}\in F,\bigcup{A_i}\in G\Rightarrow \bigcup{A_i}\in F\cap G$

Thus $F\cap G$ is a sigma-algebra.

Let

$F=\left\{ \emptyset ,\left\{ 1 \right\} ,\left\{ 2,3 \right\} ,\left\{ 1,2,3 \right\} \right\} \\G=\left\{ \emptyset ,\left\{ 2 \right\} ,\left\{ 1,3 \right\} ,\left\{ 1,2,3 \right\} \right\} \\F\cup G=\left\{ \emptyset ,\left\{ 1 \right\} ,\left\{ 2 \right\} ,\left\{ 1,3 \right\} ,\left\{ 2,3 \right\} ,\left\{ 1,2,3 \right\} \right\}$

We see that $\left\{ 1 \right\} \in F\cup G,\left\{ 2 \right\} \in F\cup G,\left\{ 1 \right\} \cup \left\{ 2 \right\} =\left\{ 1,2 \right\} \notin F\cup G$

Thus $F\cup G$ is not a sigma-algebra.