Sigma Algebra Exercise

• Aug 17th 2011, 06:07 AM
osodud
Sigma Algebra Exercise
Hello

Could you plese help me on this exercise?

Let F1 and F2 two sigma algebras of subsets of $\Omega$ and $F1\subseteq{F2}$
Prove that $F1\cup{F2}$ is a sigma algebra.
• Aug 17th 2011, 06:35 AM
girdav
Re: Sigma Algebra Exercise
Apply the definition. Show that :
(i) $\Omega\in F_1\cup F_2$;
(ii) If $A\in F_1\cup F_2$, show that $\complement_{\Omega}A\in F_1\cup F_2$;
(iii) Let $\{A_n\}_{n\in\mathbb N}\in F_1\cup F_2$. Show that $\bigcup_{n\in\mathbb N}A_n\in F_1\cup F_2$.
• Aug 17th 2011, 07:27 AM
FernandoRevilla
Re: Sigma Algebra Exercise
Another way: $F_1\cup F_2=F_2$ which by hypothesis is $\sigma$-algebra.
• Aug 17th 2011, 08:35 AM
girdav
Re: Sigma Algebra Exercise
We notice that the hypothesis $F_1\subset F_2$ (or $F_2\subset F_1$) is necessary: if we consider $\Omega=\left\{1;2;3\right\}$, $F_1 =\left\{\emptyset, \left\{1\right\},\left\{2;3\right\},\Omega\right\}$ and $F_2 =\left\{\emptyset, \left\{2\right\},\left\{1;3\right\},\Omega\right\}$ then $F_1\cup F_2 =\left\{\emptyset, \left\{1\right\},\left\{2\right\}; \left\{1;3\right\};\left\{2;3\right\},\Omega\right \}$ which is not a $\sigma$-algebra since $\left\{1\right\}\cup \left\{2\right\}\notin F_1\cup F_2$.