Hello
Could you plese help me on this exercise?
Let F1 and F2 two sigma algebras of subsets of $\displaystyle \Omega$ and $\displaystyle F1\subseteq{F2}$
Prove that $\displaystyle F1\cup{F2}$ is a sigma algebra.
Apply the definition. Show that :
(i) $\displaystyle \Omega\in F_1\cup F_2$;
(ii) If $\displaystyle A\in F_1\cup F_2$, show that $\displaystyle \complement_{\Omega}A\in F_1\cup F_2$;
(iii) Let $\displaystyle \{A_n\}_{n\in\mathbb N}\in F_1\cup F_2$. Show that $\displaystyle \bigcup_{n\in\mathbb N}A_n\in F_1\cup F_2$.
We notice that the hypothesis $\displaystyle F_1\subset F_2$ (or $\displaystyle F_2\subset F_1$) is necessary: if we consider $\displaystyle \Omega=\left\{1;2;3\right\}$, $\displaystyle F_1 =\left\{\emptyset, \left\{1\right\},\left\{2;3\right\},\Omega\right\}$ and $\displaystyle F_2 =\left\{\emptyset, \left\{2\right\},\left\{1;3\right\},\Omega\right\}$ then $\displaystyle F_1\cup F_2 =\left\{\emptyset, \left\{1\right\},\left\{2\right\}; \left\{1;3\right\};\left\{2;3\right\},\Omega\right \}$ which is not a $\displaystyle \sigma$-algebra since $\displaystyle \left\{1\right\}\cup \left\{2\right\}\notin F_1\cup F_2$.