union of sigma algebra is sigma algebra I need help to demonstrate:
Let  $F,G$  two $\sigma$-algebra in $\Omega$, if $F\cup G$ is $\sigma$-algebra in $\Omega$, show $F\subset G$ or $G\subset F$.
Thanks

Assume not. Then there exists $f\in F$ and $g\in G$ such that neither is in the other set. Since $F\cup G$ is a $\sigma$-algebra, $f\cup g$ is in it. Thus it's in at least one of $F$ and $G$. If $f\cup g$ is in $F$, then $$g=g\cap (f\cup g)\in F$$ gives a contradiction. Likewise If $f\cup g$ is in $G$, then $$f=f\cap (f\cup g)\in F$$ gives a contradiction. 
